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 1-4 of 4 results.

A328874 Constant term in the expansion of (-1 + (1 + x + 1/x) * (1 + y + 1/y) * (1 + z + 1/z))^n.

Original entry on oeis.org

1, 0, 26, 264, 5646, 101520, 2103740, 43632960, 942507790, 20685977760, 462661368876, 10483696885200, 240373512418116, 5564581640601984, 129901678525143096, 3054381796821779424, 72272856926974596750, 1719662128611006026304, 41120565854695068532076, 987633314722818034066224
Offset: 0

Views

Author

Seiichi Manyama, Oct 29 2019

Keywords

Comments

Also number of n-step closed walks (from origin to origin) in cubic lattice, using steps (t_1,t_2,t_3) (t_k = -1, 1 or 0 for 1 <= k <= 3) except for (0,0,0).

Links

Crossrefs

Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m: A126869 (m=1), A094061 (m=2), this sequence (m=3), A328875 (m=4).
Cf. A326920.

Programs

  • Mathematica
    Table[Sum[(-1)^(n-k) * Binomial[n, k] * Sum[Binomial[k, 2*j]*Binomial[2*j, j], {j, 0, k}]^3, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 30 2019 *)
Table[Sum[(-1)^(n-k) * Binomial[n, k] * Hypergeometric2F1[1/2 - k/2, -k/2, 1, 4]^3, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 30 2019 *)
  • PARI
    {a(n) = polcoef(polcoef(polcoef((-1+(1+x+1/x)*(1+y+1/y)*(1+z+1/z))^n, 0), 0), 0)}
  • PARI
    {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^3)}

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^3.
From Vaclav Kotesovec, Oct 30 2019: (Start)
Recurrence: (n-1)*n^3*(5103*n^4 - 38556*n^3 + 107838*n^2 - 132564*n + 60401)*a(n) = (n-1)^2*(71442*n^6 - 575505*n^5 + 1817613*n^4 - 2850549*n^3 + 2299999*n^2 - 891084*n + 132528)*a(n-1) + (1505385*n^8 - 17395560*n^7 + 85857516*n^6 - 235935678*n^5 + 393710399*n^4 - 407039414*n^3 + 253464484*n^2 - 86477832*n + 12324048)*a(n-2) + 2*(n-2)*(1224720*n^7 - 13539960*n^6 + 61400268*n^5 - 146649411*n^4 + 197630220*n^3 - 149760433*n^2 + 59083626*n - 9168258)*a(n-3) - 4*(n-3)*(n-2)*(1153278*n^6 - 11020212*n^5 + 40809852*n^4 - 74540514*n^3 + 70559711*n^2 - 32643654*n + 5797748)*a(n-4) - 8*(n-4)*(n-3)*(n-2)*(1367604*n^5 - 9649206*n^4 + 23421096*n^3 - 25438791*n^2 + 12638258*n - 2271566)*a(n-5) - 1040*(n-5)*(n-4)*(n-3)*(n-2)*(5103*n^4 - 18144*n^3 + 22788*n^2 - 12144*n + 2222)*a(n-6).
a(n) ~ 13 * 26^(n + 1/2) / (108 * Pi^(3/2) * n^(3/2)). (End)

A326920 Constant term in the expansion of (-1 + Product_{k=1..n} (1 + x_k + 1/x_k))^n.

Original entry on oeis.org

1, 0, 8, 264, 121200, 332810400, 7753173594200, 1440193875113407680, 2250630808138439243100640, 29565964235758317208187044137600, 3307988125501026209547184198622507128848, 3165738749695300492286911657015518806826344524560
Offset: 0

Views

Author

Seiichi Manyama, Oct 29 2019

Keywords

Comments

Also number of n-step closed walks (from origin to origin) in n-dimensional lattice, using steps (t_1,t_2, ... ,t_n) (t_k = -1, 1 or 0 for 1 <= k <= n) except for (0,0, ... ,0) (t_k = 0 for 1 <= k <= n).

Links

Crossrefs

Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m: A126869 (m=1), A094061 (m=2), A328874 (m=3), A328875 (m=4).

Programs

  • Mathematica
    Table[Sum[(-1)^(n-k) * Binomial[n, k] * Sum[Binomial[k, 2*j]*Binomial[2*j, j], {j, 0, k}]^n, {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Oct 30 2019 *)
  • PARI
    {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^n)}

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^n.
a(n) ~ 3^(n^2 + n/2) / (exp(3/16) * 2^n * Pi^(n/2) * n^(n/2)). - Vaclav Kotesovec, Oct 30 2019

A327751 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of (-1 + Product_{j=1..n} (1 + x_j + 1/x_j))^k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 8, 0, 1, 0, 6, 24, 26, 0, 1, 0, 0, 216, 264, 80, 0, 1, 0, 20, 1200, 5646, 2160, 242, 0, 1, 0, 0, 8840, 101520, 121200, 16080, 728, 0, 1, 0, 70, 58800, 2103740, 6136800, 2410326, 115464, 2186, 0, 1
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2019

Keywords

Comments

T(n,k) is the number of k-step closed walks (from origin to origin) in n-dimensional lattice, using steps (t_1,t_2, ... ,t_n) (t_j = -1, 1 or 0 for 1 <= j <= n) except for (0,0, ... ,0) (t_j = 0 for 1 <= j <= n).

Examples

			Square array begins:
   1, 0,   0,     0,       0,         0, ...
   1, 0,   2,     0,       6,         0, ...
   1, 0,   8,    24,     216,      1200, ...
   1, 0,  26,   264,    5646,    101520, ...
   1, 0,  80,  2160,  121200,   6136800, ...
   1, 0, 242, 16080, 2410326, 332810400, ...

Links

Crossrefs

Columns k=0-3 give A000012, A000004, A024023, 24*A016212(n-2).
Rows n=0-4 give A000007, A126869, A094061, A328874, A328875.
Main diagonal is A326920.
Cf. A002426, A328718.

Formula

T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * A002426(j)^n.

A329820 Triangular array, read by rows: T(n,k) = [(w*x*y*z)^k] (-1 + (1 + w + 1/w)*(1 + x + 1/x)*(1 + y + 1/y)*(1 + z + 1/z))^n for -n <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 14, 80, 14, 1, 1, 78, 1251, 2160, 1251, 78, 1, 1, 252, 9682, 60444, 121200, 60444, 9682, 252, 1, 1, 620, 49355, 760800, 3785750, 6136800, 3785750, 760800, 49355, 620, 1, 1, 1290, 190746, 5950070, 60898395, 228400980, 356570960, 228400980, 60898395, 5950070, 190746, 1290, 1
Offset: 0

Views

Author

Seiichi Manyama, Nov 21 2019

Keywords

Examples

			Triangle begins:
                          1;
                  1,      0,     1;
           1,    14,     80,    14,    1;
     1,   78,  1251,   2160,  1251,   78,   1;
1, 252, 9682, 60444, 121200, 60444, 9682, 252, 1;

Crossrefs

T(n,0) gives A328875.
Cf. A260492, A329816, A329819.

Programs

  • PARI
    {T(n, k) = polcoef(polcoef(polcoef(polcoef((-1+(1+w+1/w)*(1+x+1/x)*(1+y+1/y)*(1+z+1/z))^n, k), k), k), k)}

Formula

T(n,k) = T(n,-k).
Showing 1-4 of 4 results.

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