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.

Showing 1-6 of 6 results.

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

A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 7, 7, 1, 0, 1, 1, 15, 31, 19, 1, 0, 1, 1, 31, 115, 175, 51, 1, 0, 1, 1, 63, 391, 1255, 991, 141, 1, 0, 1, 1, 127, 1267, 8071, 13671, 5881, 393, 1, 0, 1, 1, 255, 3991, 49399, 161671, 160461, 35617, 1107, 1, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 27 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, 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, ...
   1, 1,  1,   1,     1,      1, ...
   0, 1,  3,   7,    15,     31, ...
   0, 1,  7,  31,   115,    391, ...
   0, 1, 19, 175,  1255,   8071, ...
   0, 1, 51, 991, 13671, 161671, ...

Links

Crossrefs

Columns k=0..5 give A019590(n+1), A000012, A002426, A172634, A328725, A328750.
Main diagonal gives A328811.
T(n,n+1) gives A328813.
Cf. A309010, A328748, A328807.

Programs

  • Mathematica
    T[n_, k_] := Sum[(-1)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

A328748 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

Original entry on oeis.org

1, 1, 0, 1, 0, -1, 1, 0, 0, 2, 1, 0, 2, 0, -3, 1, 0, 6, 0, 0, 4, 1, 0, 14, 12, 6, 0, -5, 1, 0, 30, 72, 90, 0, 0, 6, 1, 0, 62, 300, 882, 360, 20, 0, -7, 1, 0, 126, 1080, 6690, 8400, 2040, 0, 0, 8, 1, 0, 254, 3612, 44706, 124920, 95180, 10080, 70, 0, -9
Offset: 0

Views

Author

Seiichi Manyama, Oct 27 2019

Keywords

Comments

T(n,k) is the constant term in the expansion of (-2 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.

Examples

			Square array begins:
    1, 1, 1,   1,    1,      1, ...
    0, 0, 0,   0,    0,      0, ...
   -1, 0, 2,   6,   14,     30, ...
    2, 0, 0,  12,   72,    300, ...
   -3, 0, 6,  90,  882,   6690, ...
    4, 0, 0, 360, 8400, 124920, ...

Links

Crossrefs

Columns k=0..5 give A097141(n+1), A000007, A126869, A002898, A328735, A328751.
T(n,n+1) gives A328814.
Cf. A309010, A328747, A328807.

Programs

  • Mathematica
    T[n_, k_] := Sum[(-2)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

A328810 a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^n.

Original entry on oeis.org

1, 3, 11, 93, 2583, 260613, 99915029, 144072750195, 808177412109895, 16892305881120020613, 1388286655114683125139201, 423109739462061163278604529475, 511885816860737850466697173188711669, 2296554708428991868313593456071099604464483
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2019

Keywords

Links

Crossrefs

Main diagonal of A328807.
Cf. A167010, A328811.

Programs

  • Mathematica
    Table[Sum[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, binomial(n, i)*sum(j=0, i, binomial(i, j)^n))}

Formula

a(n) ~ A167010(n). - Vaclav Kotesovec, Oct 28 2019

A328808 Constant term in the expansion of (3 + x + y + z + 1/x + 1/y + 1/z + x*y + y*z + z*x + 1/(x*y) + 1/(y*z) + 1/(z*x) + x*y*z + 1/(x*y*z))^n.

Original entry on oeis.org

1, 3, 23, 225, 2583, 32133, 422069, 5757699, 80790775, 1158593589, 16905540753, 250185539079, 3746205581589, 56652844671855, 864032059578879, 13274539401672345, 205252378269637815, 3191578469685269925, 49876569284504593505, 782943268394316187815
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2019

Keywords

Crossrefs

Column k=4 of A328807.
Cf. A005260, A328725, A328735.

Programs

  • Mathematica
    Table[Sum[Binomial[n, i]*Sum[Binomial[i, j]^4, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2019 *)
  • PARI
    {a(n) = polcoef(polcoef(polcoef((1+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}
  • PARI
    {a(n) = sum(i=0, n, binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}

Formula

a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^3*a(n) = (2*n - 1)*(8*n^2 - 8*n + 3)*a(n-1) + (n-1)*(22*n^2 - 44*n + 13)*a(n-2) - 44*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 51*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ sqrt(2) * 17^(n + 3/2) / (64 * Pi^(3/2) * n^(3/2)). (End)

A328809 Constant term in the expansion of (1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

Original entry on oeis.org

1, 3, 39, 597, 11991, 260613, 6129489, 151078707, 3867441111, 101852866533, 2744610170049, 75348380209347, 2100889194001761, 59349600029522403, 1695505948476461559, 48909452234258070117, 1422877722974198091351, 41704912707174877940613
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2019

Keywords

Crossrefs

Column k=5 of A328807.
Cf. A005261, A328750, A328751.

Programs

  • Mathematica
    Table[Sum[Binomial[n, i]*Sum[Binomial[i, j]^5, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2019 *)
  • PARI
    {a(n) = sum(i=0, n, binomial(n, i)*sum(j=0, i, binomial(i, j)^5))}

Formula

a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^4*(40*n^2 - 24*n - 79)*a(n) = (1080*n^6 - 2808*n^5 + 875*n^4 + 2928*n^3 - 3762*n^2 + 1834*n - 336)*a(n-1) + (9320*n^6 - 42872*n^5 + 61193*n^4 - 12152*n^3 - 35518*n^2 + 21658*n - 2016)*a(n-2) - (n-2)*(48560*n^5 - 223376*n^4 + 216118*n^3 + 381866*n^2 - 791133*n + 355194)*a(n-3) + (n-3)*(n-2)*(79560*n^4 - 286416*n^3 - 56675*n^2 + 976675*n - 616322)*a(n-4) - 11*(n-4)*(n-3)*(n-2)*(5080*n^3 - 8128*n^2 - 25641*n + 21693)*a(n-5) + 363*(n-5)*(n-4)*(n-3)*(n-2)*(40*n^2 + 56*n - 63)*a(n-6).
a(n) ~ 33^(n+2) / (256 * sqrt(5) * Pi^2 * n^2). (End)
Showing 1-6 of 6 results.

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