A335310 -id:A335310 - cp's OEIS frontend

A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2(k-1)x + ((k+1)*x)^2).

1, 1, 1, 1, 0, 1, 1, -1, -2, 1, 1, -2, -3, 0, 1, 1, -3, -2, 11, 6, 1, 1, -4, 1, 28, 1, 0, 1, 1, -5, 6, 45, -74, -81, -20, 1, 1, -6, 13, 56, -255, -92, 141, 0, 1, 1, -7, 22, 55, -554, 477, 1324, 363, 70, 1, 1, -8, 33, 36, -959, 2376, 2689, -3656, -1791, 0, 1

Offset: 0

Seiichi Manyama, May 02 2019

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y) + k*x*y). - Seiichi Manyama, Jul 11 2020

More generally, column k is the diagonal of the rational function r / ((1-r*x)*(1-r*y) + r-1 + (k-r+1)*r*x*y) for any nonzero real number r. - Seiichi Manyama, Jul 22 2020

			Square array begins:
  1,   1,   1,    1,    1,    1,      1, ...
  1,   0,  -1,   -2,   -3,   -4,     -5, ...
  1,  -2,  -3,   -2,    1,    6,     13, ...
  1,   0,  11,   28,   45,   56,     55, ...
  1,   6,   1,  -74, -255, -554,   -959, ...
  1,   0, -81,  -92,  477, 2376,   6475, ...
  1, -20, 141, 1324, 2689, -804, -20195, ...

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

Columns k=2..4 give (-1)^n * A098332, A116091, (-1)^n * A098341.
Main diagonal gives A307885.
T(n,n-1) gives A335310.
Cf. A307883, A336179.

T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

T(n,k) is the coefficient of x^n in the expansion of (1 - (k-1)*x - k*x^2)^n.
T(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^2.
T(n,k) = Sum_{j=0..n} (-k-1)^(n-j) * binomial(n,j) * binomial(n+j,j).
n * T(n,k) = -(k-1) * (2*n-1) * T(n-1,k) - (k+1)^2 * (n-1) * T(n-2,k).

A335309 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^(n-k).

Original entry on oeis.org

1, 3, 22, 245, 3606, 65527, 1411404, 35066313, 985483270, 30869546411, 1065442493556, 40144438269949, 1638733865336764, 72012798200637855, 3388250516614331416, 169894851136173584145, 9041936334960057699654, 508945841697238471315027, 30202327515992972576218980

Offset: 0

Ilya Gutkovskiy, May 31 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..381

Cf. A001850, A069835, A084771, A084772, A098659, A187021, A307883, A331656, A335310.

Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] n^(n - k), {k, 0, n}], {n, 1, 18}]]
Table[SeriesCoefficient[1/Sqrt[1 - 2 (n + 2) x + n^2 x^2], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 Sqrt[n + 1] x], {x, 0, n}], {n, 0, 18}]
Table[Hypergeometric2F1[-n, -n, 1, 1 + n], {n, 0, 18}]

a(n) = sum(k=0, n, binomial(n,k)^2*(n+1)^k); \\ Michel Marcus, Jun 01 2020

a(n) = central coefficient of (1 + (n + 2)*x + (n + 1)*x^2)^n.
a(n) = [x^n] 1 / sqrt(1 - 2*(n + 2)*x + n^2*x^2).
a(n) = n! * [x^n] exp((n + 2)*x) * BesselI(0,2*sqrt(n + 1)*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * (n+1)^k.
a(n) ~ exp(2*sqrt(n)) * n^(n - 1/4) / (2*sqrt(Pi)) * (1 + 11/(12*sqrt(n))). - Vaclav Kotesovec, Jan 09 2023

cp's OEIS Frontend

A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2(k-1)x + ((k+1)*x)^2).

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A335309 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^(n-k).

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cp's OEIS Frontend

A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A335309 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^(n-k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2(k-1)x + ((k+1)*x)^2).