A357533 -id:A357533

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A122852

Row sums of number triangle A122851.

+10
9

1, 1, 2, 3, 6, 11, 24, 51, 122, 291, 756, 1979, 5526, 15627, 46496, 140451, 442194, 1414931, 4687212, 15785451, 54764846, 193129659, 698978136, 2570480147, 9672977706, 36967490691, 144232455524, 571177352091, 2304843053382, 9434493132011, 39289892366736

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Essentially the same as A072374. - R. J. Mathar, Jun 18 2008

Diagonal sums of A008279. - Paul Barry, Feb 11 2009

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Jonathan Fang, Zachary Hamaker, and Justin Troyka, On pattern avoidance in matchings and involutions, arXiv:2009.00079 [math.CO], 2020. See Theorem 1.6 (b).

Guo-Niu Han, Hankel Continued fractions and Hankel determinants of the Euler numbers, arXiv:1906.00103 [math.CO], 2019. See p. 27.

Qiong Qiong Pan and Jiang Zeng, The gamma-coefficients of Branden's (p,q)-Eulerian polynomials and André permutations, arXiv:1910.01747 [math.CO], 2019.

FORMULA

a(n) = Sum{k=0..n} C(k,n-k)*(n-k)!.

From Paul Barry, Feb 11 2009: (Start)

G.f.: 1/(1-x-x^2/(1-x^2/(1-x-2x^2/(1-2x^2/(1-x-3x^2/(1-3x^2/(1-x-4x^2/(1-4x^2/(1-... (continued fraction).

a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*k!. (End)

D-finite with recurrence -2*a(n) + 3*a(n-1) + (n-1)*a(n-2) + (-n+1)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012. Proof in [Han 2019]

a(n) ~ sqrt(Pi) * exp(sqrt(n/2) - n/2 + 1/8) * n^((n+1)/2) / 2^(n/2+1) * (1 + 37/(48*sqrt(2*n))). - Vaclav Kotesovec, Feb 08 2014

a(n) = (a(n-1) + n * a(n-2) + 1)/2 for n > 1. - Seiichi Manyama, Nov 19 2022

MATHEMATICA

Table[Sum[Binomial[n-k, k]*k!, {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 08 2014 *)

PROG

(PARI) a(n) = sum(k=0, n, binomial(k, n-k)*(n-k)!); \\ Michel Marcus, Sep 02 2020

CROSSREFS

Cf. A072374, A008279.

Cf. A357532, A357533, A357570.

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Sep 14 2006

EXTENSIONS

More terms from Vaclav Kotesovec, Jun 04 2019

STATUS

approved

A357532

a(n) = Sum_{k=0..floor(n/3)} (n-2*k)!/(n-3*k)!.

+10
6

1, 1, 1, 2, 3, 4, 7, 12, 19, 34, 63, 112, 211, 414, 799, 1588, 3267, 6706, 13999, 30024, 64723, 141142, 314271, 705724, 1599619, 3685338, 8573167, 20112016, 47804499, 114743614, 277615903, 679057092, 1676636611, 4171532674, 10477002159, 26545428568, 67755344467, 174386589606

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = (2 * a(n-1) + n * a(n-3) + 1)/3 for n > 2.

a(n) ~ c * n^(n/3 + 1/2) / (3^(n/3) * exp(n/3 - n^(2/3)/3^(2/3) - 2*n^(1/3) / 3^(7/3))) * (1 + 1235/(729 * 3^(2/3) * n^(1/3)) + 9452027/(15943230 * 3^(1/3) * n^(2/3)) + 16015315669/(41841412812*n)), where c = 0.50682110703119..., conjecture: c = exp(4/81) * sqrt(2*Pi) / 3^(3/2). - Vaclav Kotesovec, Nov 25 2022

PROG

(PARI) a(n) = sum(k=0, n\3, (n-2*k)!/(n-3*k)!);

CROSSREFS

Cf. A072374, A122852, A357533, A357570.

KEYWORD

nonn,easy

AUTHOR

Seiichi Manyama, Nov 19 2022

STATUS

approved

A357570

a(n) = Sum_{k=0..floor(n/5)} (n-4*k)!/(n-5*k)!.

+10
5

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 14, 21, 30, 41, 60, 93, 146, 225, 336, 509, 798, 1281, 2060, 3261, 5154, 8273, 13536, 22365, 36806, 60369, 99588, 166301, 280650, 474801, 802424, 1358973, 2317806, 3987185, 6893196, 11933949, 20690738, 36022161, 63107520, 111146141, 196322454, 347412753

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1000

FORMULA

D-finite with recurrence a(n) = (4 * a(n-1) + n * a(n-5) + 1)/5 for n > 4.

PROG

(PARI) a(n) = sum(k=0, n\5, (n-4*k)!/(n-5*k)!);

CROSSREFS

Cf. A072374, A122852, A357532, A357533.

KEYWORD

nonn,easy

AUTHOR

Seiichi Manyama, Nov 19 2022

STATUS

approved

A358605

a(n) = Sum_{k=0..floor(n/4)} (-1)^k * (n-3*k)!/(n-4*k)!.

+10
4

1, 1, 1, 1, 0, -1, -2, -3, -2, 1, 6, 13, 16, 9, -14, -59, -108, -119, -26, 261, 736, 1177, 1026, -731, -4964, -11079, -14978, -6299, 30024, 102841, 189466, 190917, -97004, -921191, -2301354, -3396539, -1674368, 7265241, 27311794, 53600101, 56943756, -31760903, -310594514, -809146971

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = (3 * a(n-1) - n * a(n-4) + 1)/4 for n > 3.

PROG

(PARI) a(n) = sum(k=0, n\4, (-1)^k*(n-3*k)!/(n-4*k)!);

CROSSREFS

Cf. A358603, A358604, A358606.

Cf. A357533.