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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A309010 Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 6, 8, 5, 1, 2, 10, 20, 16, 6, 1, 2, 18, 56, 70, 32, 7, 1, 2, 34, 164, 346, 252, 64, 8, 1, 2, 66, 488, 1810, 2252, 924, 128, 9, 1, 2, 130, 1460, 9826, 21252, 15184, 3432, 256, 10, 1, 2, 258, 4376, 54850, 206252, 263844, 104960, 12870, 512, 11
Offset: 0

Views

Author

Seiichi Manyama, Jul 06 2019

Keywords

Comments

A(n,k) is the constant term in the expansion of (Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0. - Seiichi Manyama, Oct 27 2019
Let B_k be the binomial poset containing all k-tuples of equinumerous subsets of {1,2,...} ordered by inclusion componentwise (described in Stanley reference below). Then A(k,n) is the number of elements in any n-interval of B_k. - Geoffrey Critzer, Apr 16 2020
Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - Product_{j=1..k} x_j) for k>0. - Seiichi Manyama, Jul 11 2020

Examples

			Square array, A(n, k), begins:
   1,  1,   1,    1,     1,      1, ... A000012;
   2,  2,   2,    2,     2,      2, ... A007395;
   3,  4,   6,   10,    18,     34, ... A052548;
   4,  8,  20,   56,   164,    488, ... A115099;
   5, 16,  70,  346,  1810,   9826, ...
   6, 32, 252, 2252, 21252, 206252, ...
Antidiagonals, T(n, k), begin:
  1;
  1,  2;
  1,  2,   3;
  1,  2,   4,    4;
  1,  2,   6,    8,    5;
  1,  2,  10,   20,   16,     6;
  1,  2,  18,   56,   70,    32,     7;
  1,  2,  34,  164,  346,   252,    64,    8;
  1,  2,  66,  488, 1810,  2252,   924,  128,   9;
  1,  2, 130, 1460, 9826, 21252, 15184, 3432, 256,  10;

References

  • R. P. Stanley, Enumerative Combinatorics Vol I, Second Edition, Cambridge, 2011, Example 3.18.3 d, page 366.

Links

Crossrefs

Columns k=0..12 give A000027(n+1), A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
Main diagonal gives A167010.
Cf. A328747, A328748, A328807, A328812.
Cf. A000012, A007395, A052548, A115099.

Programs

  • Magma
    [(&+[Binomial(k,j)^(n-k): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 26 2022
  • Mathematica
    nn = 8; Table[ek[x_] := Sum[x^n/n!^k, {n, 0, nn}];Range[0, nn]!^k CoefficientList[Series[ek[x]^2, {x, 0, nn}],x], {k, 0, nn}] // Transpose // Grid (* Geoffrey Critzer, Apr 17 2020 *)
  • PARI
    A(n, k) = sum(j=0, n, binomial(n, j)^k); \\ Seiichi Manyama, Jan 08 2022
  • SageMath
    flatten([[sum(binomial(k,j)^(n-k) for j in (0..k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 26 2022

Formula

A(n, k) = Sum_{j=0..n} binomial(n,j)^k (array).
A(n, n+1) = A328812(n).
A(n, n) = A167010(n).
T(n, k) = A(k, n-k) (antidiagonals).
T(n, n) = A000027(n+1).
T(n, n-1) = A000079(n-1).
T(n, n-2) = A000984(n-2).
T(n, n-3) = A000172(n-3).
T(n, n-4) = A005260(n-4).
T(n, n-5) = A005261(n-5).
T(n, n-6) = A069865(n-6).
T(n, n-7) = A182421(n-7).
T(n, n-8) = A182422(n-8).
T(n, n-9) = A182446(n-9).
T(n, n-10) = A182447(n-10).
T(n, n-11) = A342294(n-11).
T(n, n-12) = A342295(n-12).
Sum_{n>=0} A(n,k) x^n/(n!^k) = (Sum_{n>=0} x^n/(n!^k))^2. - Geoffrey Critzer, Apr 17 2020

A336214 a(n) = Sum_{k=0..n} k^n * binomial(n,k)^n, with a(0)=1.

Original entry on oeis.org

1, 1, 8, 270, 41984, 30706250, 94770093312, 1336016204844832, 76829717664330940416, 19838680914222199482800274, 20521247958509575370600000000000, 94285013320530947020636486516362047300, 1715947732437668013396578734960052732361179136
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 12 2020

Keywords

Links

Crossrefs

Cf. A072034, A086331, A167008, A167010, A328812, A336188, A336204, A336212, A336213.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[k^n*Binomial[n, k]^n, {k, 1, n}], {n, 1, 15}]}]
  • PARI
    a(n) = if (n==0, 1, sum(k=0, n, k^n * binomial(n,k)^n)); \\ Michel Marcus, Jul 13 2020

Formula

a(n) ~ c * exp(-1/4) * 2^(n^2 - n/2) * n^(n/2) / Pi^(n/2), where c = Sum_{k = -infinity..infinity} exp(-2*k*(k-1)) = exp(1/2) * sqrt(Pi/2) * EllipticTheta(3, -Pi/2, exp(-Pi^2/2)) = 2.036643566277677716389243890291939003151565... if n is even and c = Sum_{k = -infinity..infinity} exp(-2*k^2 + 1/2) = exp(1/2) * EllipticTheta(3, 0, exp(-2)) = 2.096087809957308346119920713317351288828811... if n is odd.
a(n) = n^n * A328812(n-1) for n > 0. - Seiichi Manyama, Jul 15 2020

A328813 Constant term in the expansion of (-1 + Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

Original entry on oeis.org

1, 1, 7, 115, 8071, 1770951, 1505946121, 4368457532265, 49949721645153751, 2021436054924485283799, 327902645022367779788597977, 191573267131797606250658812550565, 453516825886934673673734108656254582801
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2019

Keywords

Links

Crossrefs

Cf. A328747, A328812, A328814.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^(n+1), {j, 0, i}], {i, 0, n}]; Array[a, 13, 0] (* Amiram Eldar, May 06 2021 *)
  • PARI
    {a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^(n+1)))}

Formula

a(n) = A328747(n,n+1) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^(n+1).

A328814 Constant term in the expansion of (-2 + Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

Original entry on oeis.org

1, 0, 6, 72, 6690, 1536000, 1398496680, 4165565871600, 48724656010825410, 1991141239554487077120, 325362786100184356140612996, 190695111051826003327799496771600, 452459020719698368348441955010421696800
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2019

Keywords

Links

Crossrefs

Cf. A328748, A328812, A328813.

Programs

  • Mathematica
    a[n_] := Sum[(-2)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^(n+1), {j, 0, i}], {i, 0, n}]; Array[a, 13, 0] (* Amiram Eldar, May 06 2021 *)
  • PARI
    {a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^(n+1)))}

Formula

a(n) = A328748(n,n+1) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^(n+1).
Showing 1-4 of 4 results.

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