cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A328807 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

Original entry on oeis.org

1, 1, 3, 1, 3, 8, 1, 3, 9, 20, 1, 3, 11, 27, 48, 1, 3, 15, 45, 81, 112, 1, 3, 23, 93, 195, 243, 256, 1, 3, 39, 225, 639, 873, 729, 576, 1, 3, 71, 597, 2583, 4653, 3989, 2187, 1280, 1, 3, 135, 1665, 11991, 32133, 35169, 18483, 6561, 2816
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2019

Keywords

Comments

T(n,k) is the constant term in the expansion of (1 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.
For fixed k > 0 is T(n,k) ~ (2^k + 1)^(n + (k-1)/2) / (2^((k-1)^2/2) * sqrt(k) * (Pi*n)^((k-1)/2)). - Vaclav Kotesovec, Oct 28 2019

Examples

			Square array begins:
     1,   1,   1,    1,     1,      1, ...
     3,   3,   3,    3,     3,      3, ...
     8,   9,  11,   15,    23,     39, ...
    20,  27,  45,   93,   225,    597, ...
    48,  81, 195,  639,  2583,  11991, ...
   112, 243, 873, 4653, 32133, 260613, ...

Links

Crossrefs

Columns k=0..5 give A001792, A000244, A026375, A002893, A328808, A328809.
Main diagonal gives A328810.
Cf. A309010, A328747, A328748.

Programs

  • Mathematica
    T[n_, k_] := Sum[Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

A328811 a(n) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^n.

Original entry on oeis.org

1, 1, 3, 31, 1255, 161671, 75481581, 121338954577, 734884394666535, 15970479086714049751, 1347242827078365957146473, 415839472158527880691583531617, 507266883682599825619985300960971525, 2284735689605775548174387143718048664963601
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2019

Keywords

Links

Crossrefs

Main diagonal of A328747.
Cf. A167010, A328810.

Programs

  • Mathematica
    Table[Sum[(-1)^(n-i)*Binomial[n, i]*Sum[Binomial[i, j]^n, {j, 0, i}], {i, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 28 2019 *)
  • PARI
    {a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^n))}

Formula

a(n) ~ A167010(n). - Vaclav Kotesovec, Oct 28 2019
Showing 1-2 of 2 results.

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