A331793 -id:A331793 - 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)

A387307 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k+2) * binomial(2*k+4,k+4).

Original entry on oeis.org

1, 15, 166, 1650, 15615, 144025, 1309084, 11793780, 105663885, 943298675, 8401596258, 74716379270, 663813240363, 5894026429725, 52314876771960, 464261939106600, 4119843554861913, 36560929542771735, 324489293583792990, 2880380080564191450, 25572856871556696471

Offset: 0

Seiichi Manyama, Aug 25 2025

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

Cf. A331793, A387308.

[&+[2^k * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025

Table[Sum[2^k*Binomial[n+2,k+2]*Binomial[2*k+4,k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)

a(n) = sum(k=0, n, 2^k*binomial(n+2, k+2)*binomial(2*k+4, k+4));

n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - 9*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+5*x+4*x^2)^(n+2).
E.g.f.: exp(5*x) * BesselI(2, 4*x) / 4, with offset 2.

A387308 a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k+3) * binomial(2*k+6,k+6).

Original entry on oeis.org

1, 20, 270, 3100, 32711, 328440, 3195276, 30433800, 285604605, 2651696300, 24425110138, 223636254660, 2038173128355, 18508562948400, 167596683043032, 1514149108652880, 13654464563944377, 122951462526317700, 1105768912324277670, 9934852156019798700, 89186900539764803391

Offset: 0

Seiichi Manyama, Aug 25 2025

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

Cf. A331793, A387307.

[&+[2^k * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025

Table[Sum[2^k*Binomial[n+3,k+3]*Binomial[2*k+6,k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)

a(n) = sum(k=0, n, 2^k*binomial(n+3, k+3)*binomial(2*k+6, k+6));

n*(n+6)*a(n) = (n+3) * (5*(2*n+5)*a(n-1) - 9*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+5*x+4*x^2)^(n+3).
E.g.f.: exp(5*x) * BesselI(3, 4*x) / 8, with offset 3.

A387309 a(n) = Sum_{k=0..n} 3^k * binomial(n+1,k+1) * binomial(2*k+2,k+2).

Original entry on oeis.org

1, 14, 174, 2128, 26045, 320082, 3951493, 48987848, 609592347, 7610525650, 95287524332, 1196054790168, 15046318739803, 189654839753750, 2394743468261190, 30285593026553536, 383554551776056139, 4863775493104574634, 61748210178809072722, 784757334938247965840, 9983152795673915802399

Offset: 0

Seiichi Manyama, Aug 25 2025

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

Cf. A387310, A387311.
Cf. A026376, A331793.
Cf. A385716.

[&+[3^k * Binomial(n+1,k+1) * Binomial(2*k+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025

Table[Sum[3^k*Binomial[n+1,k+1]*Binomial[2*k+2,k+2],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)

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

G.f.: ((1-7*x)/sqrt((1-x) * (1-13*x)) - 1)/(18*x^2).
n*(n+2)*a(n) = (n+1) * (7*(2*n+1)*a(n-1) - 13*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+7*x+9*x^2)^(n+1).
E.g.f.: exp(7*x) * BesselI(1, 6*x) / 3, 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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A387307 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k+2) * binomial(2*k+4,k+4).

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A387308 a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k+3) * binomial(2*k+6,k+6).

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A387309 a(n) = Sum_{k=0..n} 3^k * binomial(n+1,k+1) * binomial(2*k+2,k+2).

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Magma

Mathematica

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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)).