The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A337617 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A337617

T(n, k) = (n + 1)*2^(n + k)*hypergeom([-n, k - n + 1], [2], 1/2), triangle read by rows for 0 <= k <= n.
(history; published version)

#16 by Peter Luschny at Tue Jan 09 14:00:23 EST 2024

STATUS

editing

approved

#15 by Peter Luschny at Tue Jan 09 14:00:07 EST 2024

FORMULA

T(n, k) = if (n = k) then 2^n*(2^(n+1)-1) else , otherwise 2^(2*k+1)*Sum_{j,=0..n-k} ((-1)^j*2^(n-k-j)*binomial(n+1,j)*binomial(2*n-j-k,n)). - Detlef Meya, Dec 20 2023

MAPLE

T := (n, k) -> simplify((n + 1)*2^(n + k)*hypergeom([-n, k - n + 1], [2], 1/2)): seq(seq(T(n, k), k=0..n), n=0..8);

seq(seq(T(n, k), k=0..n), n=0..8);

MATHEMATICA

T[n_, k_] := If[n==k, 2^n*(2^(n+1)-1), 2^(2*k+1)*Sum[(-1)^j*2^(n-k-j)* Binomial[n+1, j]*Binomial[2*n-j-k, n], {j, 0, n-k}]]; Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* _Detlef Meya_, Dec 20 2023 *)

Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Dec 20 2023 *)

STATUS

approved

editing

Discussion

Tue Jan 09

14:00

Peter Luschny: Format only.

#14 by Michael De Vlieger at Wed Dec 20 12:36:21 EST 2023

STATUS

reviewed

approved

#13 by Peter Luschny at Wed Dec 20 11:37:38 EST 2023

STATUS

proposed

reviewed

#12 by Michel Marcus at Wed Dec 20 11:17:14 EST 2023

STATUS

editing

proposed

Discussion

Wed Dec 20

11:37

Peter Luschny: The main diagonal of the triangle looks strange. Perhaps the author wanted to calculate this sum: def T(n, k): return sum((-1)^j*2**(n+k+1-j)*binomial(n+1, j)*binomial(2*n-k-j, n) for j in range(n-k+1)) and use this hyper: def T(n, k): return 2**(k + n + 1)*binomial(2*n - k, n)* hypergeometric([-n - 1, k - n], [k - 2*n], 1/2). Never mind.

#11 by Michel Marcus at Wed Dec 20 11:17:09 EST 2023

FORMULA

T(n,k) = if (n=k) 2^n*(2^(n+1)-1) else 2^(2*k+1)*Sum_{j,0..n-k}_ ((-1)^j*2^(n-k-j)*binomial(n+1,j)*binomial(2*n-j-k,n)). - Detlef Meya, Dec 20 2023

STATUS

proposed

editing

#10 by Detlef Meya at Wed Dec 20 10:58:46 EST 2023

STATUS

editing

proposed

#9 by Detlef Meya at Wed Dec 20 10:58:42 EST 2023

FORMULA

T(n,k) = if (n=k) 2^n*(2^(n+1)-1) else 2^(2*k+1)*Sum{j,0..n-k}_((-1)^j*2^(n-k-j)*binomial(n+1,j)*binomial(2*n-j-k,n)). - Detlef Meya, Dec 20 2023

MATHEMATICA

T[n_, k_]:=If[n==k, 2^n*(2^(n+1)-1), 2^(2*k+1)*Sum[(-1)^j*2^(n-k-j)*Binomial[n+1, j]*Binomial[2*n-j-k, n], {j, 0, n-k}]]; Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Dec 20 2023 *)

STATUS

approved

editing

#8 by Joerg Arndt at Mon Mar 08 02:14:39 EST 2021

STATUS

reviewed

approved

#7 by Michel Marcus at Sun Mar 07 23:39:47 EST 2021

STATUS

proposed

reviewed