A331792 -id:A331792 - cp's OEIS frontend

A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2kx + ((k-2)x)^2 + (1 - kx) * sqrt(1 - 2kx + ((k-2)*x)^2)).

1, 1, 0, 1, 2, -3, 1, 4, 3, 0, 1, 6, 15, 4, 10, 1, 8, 33, 56, 5, 0, 1, 10, 57, 180, 210, 6, -35, 1, 12, 87, 400, 985, 792, 7, 0, 1, 14, 123, 740, 2810, 5418, 3003, 8, 126, 1, 16, 165, 1224, 6285, 19824, 29953, 11440, 9, 0, 1, 18, 213, 1876, 12130, 53550, 140497, 166344, 43758, 10, -462

Offset: 0

Seiichi Manyama, Jan 26 2020

			Square array begins:
   1, 1,   1,    1,     1,     1, ...
   0, 2,   4,    6,     8,    10, ...
  -3, 3,  15,   33,    57,    87, ...
   0, 4,  56,  180,   400,   740, ...
  10, 5, 210,  985,  2810,  6285, ...
   0, 6, 792, 5418, 19824, 53550, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..5 give A000027(n+1), A001791(n+1), A050151(n+1), A331792, A331793.
T(n,n+1) gives A331794.
Cf. A307883, A331511, A331514, A331795.

T[n_, k_] := Sum[If[k==1 && j==0, 1, (k-1)^j] * Binomial[n + 1, j] * Binomial[n + 1, j + 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 05 2021 *)

T(n,k) = Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
n * (n+2) * T(n,k) = (n+1) * (k * (2*n+1) * T(n-1,k) - (k-2)^2 * n * T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (k-1)^j * k^(n-2*j) * binomial(n+1,n-2*j) * binomial(2*j+1,j). - Seiichi Manyama, Aug 24 2025
From Seiichi Manyama, Aug 27 2025: (Start)
T(n,k) = [x^n] (1+k*x+(k-1)*x^2)^(n+1).
For k != 1, e.g.f. of column k: exp(k*x) * BesselI(1, 2*sqrt(k-1)*x) / sqrt(k-1), with offset 1. (End)

A387339 a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k) * binomial(n+2,k+2).

Original entry on oeis.org

1, 12, 108, 880, 6855, 52164, 391720, 2918304, 21634290, 159880600, 1179180552, 8685874080, 63930198787, 470327654580, 3459353475600, 25442360389696, 187126561024686, 1376455855989672, 10126540146288520, 74515694338112160, 548444877468906726

Offset: 0

Seiichi Manyama, Aug 27 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A069835, A331792, A387340.

Cf. A387310.

[&+[3^k * Binomial(n+2,k) * Binomial(n+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Table[Sum[3^k * Binomial[n+2,k]*Binomial[n+2, k+2],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)

a(n) = sum(k=0, n, 3^k*binomial(n+2, k)*binomial(n+2, k+2));

n*(n+4)*a(n) = (n+2) * (4*(2*n+3)*a(n-1) - 4*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+4*x+3*x^2)^(n+2).
E.g.f.: exp(4*x) * BesselI(2, 2*sqrt(3)*x) / 3, with offset 2.

A387340 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).

Original entry on oeis.org

1, 16, 175, 1640, 14189, 117152, 939036, 7379040, 57188010, 438810592, 3342302821, 25316084248, 190937278805, 1435287936320, 10760879892008, 80509920297792, 601343784616830, 4485466826475360, 33420579148668670, 248788060638391120, 1850652536242372786

Offset: 0

Seiichi Manyama, Aug 27 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A069835, A331792, A387339.

Cf. A002696, A387338.

[&+[3^k * Binomial(n+3,k) * Binomial(n+3,k+3): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Table[Sum[3^k * Binomial[n+3,k]*Binomial[n+3, k+3],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)

a(n) = sum(k=0, n, 3^k*binomial(n+3, k)*binomial(n+3, k+3));

n*(n+6)*a(n) = (n+3) * (4*(2*n+5)*a(n-1) - 4*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+4*x+3*x^2)^(n+3).
E.g.f.: exp(4*x) * BesselI(3, 2*sqrt(3)*x) / (3*sqrt(3)), with offset 3.

A387368 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+1,k) * binomial(n+1,n-k).

Original entry on oeis.org

1, 10, 93, 860, 7985, 74550, 699685, 6597400, 62457921, 593346050, 5653702637, 54012503220, 517192500721, 4962377183470, 47698928343285, 459224987322800, 4427611044899585, 42744433267222650, 413145666547033213, 3997556929553596300, 38718094094951086641

Offset: 0

Seiichi Manyama, Aug 27 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A006442, A387369.

Cf. A331792.

[&+[2^k * 3^(n-k) * Binomial(n+1,k) * Binomial(n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Table[Sum[2^k * 3^(n-k)*Binomial[n+1,k]*Binomial[n+1, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)

a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(n+1, k)*binomial(n+1, n-k));

a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(n+1,k) * binomial(n+1,n-k).
n*(n+2)*a(n) = (n+1) * (5*(2*n+1)*a(n-1) - n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 6^k * 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+5*x+6*x^2)^(n+1).
E.g.f.: exp(5*x) * BesselI(1, 2*sqrt(6)*x) / sqrt(6), with offset 1.

cp's OEIS Frontend

A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2*k*x + ((k-2)*x)^2 + (1 - k*x) * sqrt(1 - 2*k*x + ((k-2)*x)^2)).

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A387339 a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k) * binomial(n+2,k+2).

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A387340 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).

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A387368 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+1,k) * binomial(n+1,n-k).

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A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2kx + ((k-2)x)^2 + (1 - kx) * sqrt(1 - 2kx + ((k-2)*x)^2)).