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O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 104*x^3 + 1772*x^4 + 42408*x^5 + 1303504*x^6 + 48736000*x^7 + 2139552016*x^8 + 107629121888*x^9 + 6094743943584*x^10 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp(n*(n+1)*x) / A(x) begins:

n=0: [1, -2, -12, -432, -32640, -4176000, -804504960, -216834831360, ...];

n=1: [1, 0, -16, -520, -36432, -4520768, -856647680, -228458074752, ...];

n=2: [1, 4, 0, -648, -46032, -5341824, -974612736, -254049782400, ...];

n=3: [1, 10, 84, 0, -56832, -6922368, -1194341760, -299397745152, ...];

n=4: [1, 18, 308, 4448, 0, -8528000, -1573784960, -376524725760, ...];

n=5: [1, 28, 768, 20088, 444720, 0, -1938504960, -502258872960, ...];

n=6: [1, 40, 1584, 61560, 2286768, 72032832, 0, -618983309952, ...];

n=7: [1, 54, 2900, 154352, 8074368, 404450176, 17201640064, 0, ...];

n=8: [1, 70, 4884, 339120, 23357568, 1583068032, 102886277760, 5682964174848, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+1)*x ) / A(x) = 0 for n>=0.

LOGARITHMIC DERIVATIVE.

The logarithmic derivative of A(x) yields the o.g.f. of A304317:

A'(x)/A(x) = 2 + 16*x + 260*x^2 + 6200*x^3 + 191832*x^4 + 7235152*x^5 + 320372320*x^6 + 16243028896*x^7 + 926219213216*x^8 + 58608051937536*x^9 + 4072302306624576*x^10 + ...+ A304317(n)*x^n + ...

O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 104*x^3 + 1772*x^4 + 42408*x^5 + 1303504*x^6 + 48736000*x^7 + 2139552016*x^8 + 107629121888*x^9 + 6094743943584*x^10 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp(n*(n+1)*x) / A(x) begins:

n=0: [1, -2, -12, -432, -32640, -4176000, -804504960, -216834831360, ...];

n=1: [1, 0, -16, -520, -36432, -4520768, -856647680, -228458074752, ...];

n=2: [1, 4, 0, -648, -46032, -5341824, -974612736, -254049782400, ...];

n=3: [1, 10, 84, 0, -56832, -6922368, -1194341760, -299397745152, ...];

n=4: [1, 18, 308, 4448, 0, -8528000, -1573784960, -376524725760, ...];

n=5: [1, 28, 768, 20088, 444720, 0, -1938504960, -502258872960, ...];

n=6: [1, 40, 1584, 61560, 2286768, 72032832, 0, -618983309952, ...];

n=7: [1, 54, 2900, 154352, 8074368, 404450176, 17201640064, 0, ...];

n=8: [1, 70, 4884, 339120, 23357568, 1583068032, 102886277760, 5682964174848, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+1)*x ) / A(x) = 0 for n>=0.

LOGARITHMIC DERIVATIVE.

The logarithmic derivative of A(x) yields the o.g.f. of A304317:

A'(x)/A(x) = 2 + 16*x + 260*x^2 + 6200*x^3 + 191832*x^4 + 7235152*x^5 + 320372320*x^6 + 16243028896*x^7 + 926219213216*x^8 + 58608051937536*x^9 + 4072302306624576*x^10 + ...+ A304317(n)*x^n + ...

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a(n) ~ sqrt(1-c) * 2^(2*n) * n^(n - 1/2) / (sqrt(Pi) * c^(n + 1/2) * (2-c)^n * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

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Cf. A304317, A304318, A304320, A304863, A304864, A304865.

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The table of coefficients of x^k /k! in exp(n*(n+1)*x) / A(x) begins:

n=0: [1, -2, -6, 12, -72, 432, -1360, 32640, -34800, 4176000, -1117368, 804504960, -43022784], 216834831360, ...];

n=1: [1, 0, -8, 16, -520, -260/3, 36432, -1518, 4520768, -565096/15, 856647680, -10708096/9, 228458074752, ...];

n=2: [1, 4, 0, -108, 648, -1918, 46032, -222576/5, 5341824, -6768144/5, 974612736, -352846920/7], 254049782400, ...];

n=3: [1, 10, 42, 84, 0, -2368, 56832, -288432/5, 6922368, -1658808, 1194341760, -2079151008/35], 299397745152, ...];

n=4: [1, 18, 154, 2224/3, 308, 4448, 0, -213200/3, 8528000, -19672312/9, 1573784960, -522951008/7], 376524725760, ...];

n=5: [1, 28, 384, 3348, 18530, 768, 20088, 444720, 0, -2692368, 1938504960, -697581768/7], 502258872960, ...];

n=6: [1, 40, 792, 10260, 95282, 3001368/5, 1584, 61560, 2286768, 72032832, 0, -614070744/5], 618983309952, ...];

n=7: [1, 54, 1450, 77176/3, 336432, 50556272/15, 1075102504/45, 2900, 154352, 8074368, 404450176, 17201640064, 0], , ...]; ...

n=8: [1, 70, 4884, 339120, 23357568, 1583068032, 102886277760, 5682964174848, 0, ...]; ...

in which the main diagonal is all zeros after the inital initial term, illustrating that [x^n] exp( n*(n+1)*x ) / A(x) = 0 for n>=0.

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The logarithmic derivative of A(x) yields the o.g.f. of A304318A304317:

A'(x)/A(x) = 2 + 16*x + 260*x^2 + 6200*x^3 + 191832*x^4 + 7235152*x^5 + 320372320*x^6 + 16243028896*x^7 + 926219213216*x^8 + 58608051937536*x^9 + 4072302306624576*x^10 + ...+ A304318A304317(n)*x^n + ...

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