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.

A361834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * binomial(2*j,j) * binomial(k*j,n-j).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 4, 20, 1, 2, 2, 8, 70, 1, 2, 0, -2, 16, 252, 1, 2, -2, -10, -14, 32, 924, 1, 2, -4, -16, -22, -32, 64, 3432, 1, 2, -6, -20, -10, 12, -30, 128, 12870, 1, 2, -8, -22, 20, 118, 174, 64, 256, 48620, 1, 2, -10, -22, 66, 242, 304, 344, 346, 512, 184756
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Examples

			Square array begins:
    1,  1,   1,   1,   1,   1,    1, ...
    2,  2,   2,   2,   2,   2,    2, ...
    6,  4,   2,   0,  -2,  -4,   -6, ...
   20,  8,  -2, -10, -16, -20,  -22, ...
   70, 16, -14, -22, -10,  20,   66, ...
  252, 32, -32,  12, 118, 242,  342, ...
  924, 64, -30, 174, 304,  82, -678, ...

Crossrefs

Columns k=0..4 give A000984, A000079, A361815, A361816, A361817.
Main diagonal gives A361835.
Cf. A361830.

Programs

  • PARI
    T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(2*j, j)*binomial(k*j, n-j));

Formula

G.f. of column k: 1/sqrt(1 - 4*x*(1-x)^k).
n*T(n,k) = 2 * Sum_{j=0..k} (-1)^j * binomial(k,j)*(2*n-1-j)*T(n-1-j,k) for n > k.

A361829 a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(n*k,n-k).

Original entry on oeis.org

1, 2, 10, 62, 486, 4482, 47106, 553226, 7152438, 100644194, 1527758136, 24839853326, 430045385424, 7888706328934, 152685931935634, 3106864307092950, 66253232332628166, 1476558925897693698, 34307420366092350048, 829217371825336147142
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Crossrefs

Main diagonal of A361830.
Cf. A099237, A361835.

Programs

  • Mathematica
    Table[Sum[Binomial[2*k,k]*Binomial[n*k,n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 26 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(2*k, k)*binomial(n*k, n-k));

Formula

a(n) = [x^n] 1/sqrt(1 - 4*x*(1+x)^n).
log(a(n)) ~ n*(log(n) + (2*log(2) - 1)/log(n) - (1 - 1/log(n))*log(log(n) - 1)). - Vaclav Kotesovec, Mar 26 2023

A361847 a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(n*k,n-k).

Original entry on oeis.org

1, 3, 12, 27, -75, -444, 4734, 11532, -466782, 1626750, 50347410, -708889296, -2196754992, 179878246239, -1795732735128, -24691325878980, 953903679982809, -7684914725016600, -226465559200630566, 7742131606464606525, -58889021552013912990
Offset: 0

Views

Author

Seiichi Manyama, Mar 27 2023

Keywords

Links

Crossrefs

Main diagonal of A361840.
Cf. A361835, A361846.

Programs

  • PARI
    a(n) = (-1)^n*sum(k=0, n, 9^k*binomial(-1/3, k)*binomial(n*k, n-k));

Formula

a(n) = [x^n] 1/(1 - 9*x*(1-x)^n)^(1/3).

A361836 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n*k,n-k).

Original entry on oeis.org

1, 1, -1, -2, 13, -29, -80, 1268, -7351, 13276, 245746, -3632793, 27451743, -63909390, -1752952501, 34899085656, -370619158447, 1779155624299, 23668687715473, -780307293795152, 12058261763444876, -107734052276914986, -180664717708949253, 30298196609011736398
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Crossrefs

Cf. A099237, A361835.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n*k, n-k));

Formula

a(n) = [x^n] 1/(1 - x*(1-x)^n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.