A381566 -id:A381566 - cp's OEIS frontend

A384581 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A143501.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 4, 0, 1, 4, 6, 10, 16, 0, 1, 5, 10, 19, 41, 92, 0, 1, 6, 15, 32, 78, 224, 616, 0, 1, 7, 21, 50, 131, 411, 1464, 4729, 0, 1, 8, 28, 74, 205, 672, 2617, 11002, 40776, 0, 1, 9, 36, 105, 306, 1031, 4170, 19251, 93234, 388057, 0

Offset: 0

Seiichi Manyama, Jun 04 2025

			Square array begins:
  1,   1,    1,    1,    1,    1,    1, ...
  0,   1,    2,    3,    4,    5,    6, ...
  0,   1,    3,    6,   10,   15,   21, ...
  0,   4,   10,   19,   32,   50,   74, ...
  0,  16,   41,   78,  131,  205,  306, ...
  0,  92,  224,  411,  672, 1031, 1518, ...
  0, 616, 1464, 2617, 4170, 6245, 8997, ...

Columns k=0..1 give A000007, A143501.

Cf. A381566, A384580, A384582, A384583.

a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, j)));

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(3*n-3*j+k,j)/(3*n-3*j+k) * A(n-j,j).

A384582 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384574.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 5, 0, 1, 4, 6, 12, 23, 0, 1, 5, 10, 22, 57, 155, 0, 1, 6, 15, 36, 105, 366, 1236, 0, 1, 7, 21, 55, 171, 651, 2853, 11286, 0, 1, 8, 28, 80, 260, 1032, 4951, 25584, 116333, 0, 1, 9, 36, 112, 378, 1536, 7656, 43587, 259789, 1329433, 0

Offset: 0

Seiichi Manyama, Jun 04 2025

			Square array begins:
  1,    1,    1,    1,    1,     1,     1, ...
  0,    1,    2,    3,    4,     5,     6, ...
  0,    1,    3,    6,   10,    15,    21, ...
  0,    5,   12,   22,   36,    55,    80, ...
  0,   23,   57,  105,  171,   260,   378, ...
  0,  155,  366,  651, 1032,  1536,  2196, ...
  0, 1236, 2853, 4951, 7656, 11125, 15552, ...

Columns k=0..1 give A000007, A384574.

Cf. A381566, A384580, A384581, A384583.

a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-4*j+k, j)/(4*n-4*j+k)*a(n-j, j)));

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(4*n-4*j+k,j)/(4*n-4*j+k) * A(n-j,j).

A384583 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384575.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 6, 0, 1, 4, 6, 14, 31, 0, 1, 5, 10, 25, 75, 236, 0, 1, 6, 15, 40, 135, 546, 2166, 0, 1, 7, 21, 60, 215, 951, 4902, 22722, 0, 1, 8, 28, 86, 320, 1476, 8338, 50620, 269889, 0, 1, 9, 36, 119, 456, 2151, 12634, 84714, 593347, 3567412, 0

Offset: 0

Seiichi Manyama, Jun 04 2025

			Square array begins:
  1,    1,    1,    1,     1,     1,     1, ...
  0,    1,    2,    3,     4,     5,     6, ...
  0,    1,    3,    6,    10,    15,    21, ...
  0,    6,   14,   25,    40,    60,    86, ...
  0,   31,   75,  135,   215,   320,   456, ...
  0,  236,  546,  951,  1476,  2151,  3012, ...
  0, 2166, 4902, 8338, 12634, 17985, 24627, ...

Columns k=0..1 give A000007, A384575.

Cf. A381566, A384580, A384581, A384582.

a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n-5*j+k, j)/(5*n-5*j+k)*a(n-j, j)));

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(5*n-5*j+k,j)/(5*n-5*j+k) * A(n-j,j).

A384580 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A143500.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 8, 10, 0, 1, 5, 10, 16, 27, 46, 0, 1, 6, 15, 28, 54, 118, 244, 0, 1, 7, 21, 45, 95, 228, 609, 1481, 0, 1, 8, 28, 68, 155, 392, 1144, 3602, 10020, 0, 1, 9, 36, 98, 240, 631, 1916, 6597, 23866, 74400, 0

Offset: 0

Seiichi Manyama, Jun 04 2025

			Square array begins:
  1,   1,   1,    1,    1,    1,    1, ...
  0,   1,   2,    3,    4,    5,    6, ...
  0,   1,   3,    6,   10,   15,   21, ...
  0,   3,   8,   16,   28,   45,   68, ...
  0,  10,  27,   54,   95,  155,  240, ...
  0,  46, 118,  228,  392,  631,  972, ...
  0, 244, 609, 1144, 1916, 3015, 4560, ...

Columns k=0..2 give A000007, A143500, A384576.

Cf. A381566, A384581, A384582, A384583.

a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*j+k)*a(n-j, j)));

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(2*n-2*j+k,j)/(2*n-2*j+k) * A(n-j,j).

A381567 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A381568.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 6, 14, 22, 0, 1, 8, 27, 64, 126, 0, 1, 10, 44, 134, 365, 884, 0, 1, 12, 65, 240, 777, 2492, 7149, 0, 1, 14, 90, 390, 1438, 5238, 19578, 64688, 0, 1, 16, 119, 592, 2440, 9696, 40244, 172356, 641836, 0, 1, 18, 152, 854, 3891, 16632, 73408, 345726, 1668686, 6888740, 0

Offset: 0

Seiichi Manyama, Feb 28 2025

			Square array begins:
  1,    1,     1,     1,     1,      1,      1, ...
  0,    2,     4,     6,     8,     10,     12, ...
  0,    5,    14,    27,    44,     65,     90, ...
  0,   22,    64,   134,   240,    390,    592, ...
  0,  126,   365,   777,  1438,   2440,   3891, ...
  0,  884,  2492,  5238,  9696,  16632,  27036, ...
  0, 7149, 19578, 40244, 73408, 125035, 203258, ...

Columns k=0..1 give A000007, A381568.

Cf. A381566, A381569.

a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+2*k, j)/(n-j+k)*a(n-j, j)));

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(2*n-2*j+2*k,j)/(n-j+k) * A(n-j,j).

A381569 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A381570.

Original entry on oeis.org

1, 1, 0, 1, 3, 0, 1, 6, 12, 0, 1, 9, 33, 82, 0, 1, 12, 63, 236, 732, 0, 1, 15, 102, 489, 2100, 7944, 0, 1, 18, 150, 868, 4428, 22248, 99156, 0, 1, 21, 207, 1400, 8121, 46422, 270268, 1381464, 0, 1, 24, 273, 2112, 13665, 85272, 552540, 3668568, 21065853, 0

Offset: 0

Seiichi Manyama, Feb 28 2025

			Square array begins:
  1,    1,     1,     1,     1,      1, ...
  0,    3,     6,     9,    12,     15, ...
  0,   12,    33,    63,   102,    150, ...
  0,   82,   236,   489,   868,   1400, ...
  0,  732,  2100,  4428,  8121,  13665, ...
  0, 7944, 22248, 46422, 85272, 145143, ...

Columns k=0..1 give A000007, A381570.

Cf. A381566, A381567.

a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+3*k, j)/(n-j+k)*a(n-j, j)));

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(3*n-3*j+3*k,j)/(n-j+k) * A(n-j,j).

cp's OEIS Frontend

A384581 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A143501.

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A384582 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384574.

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A384583 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384575.

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A384580 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A143500.

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A381567 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A381568.

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A381569 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A381570.

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