A379599 -id:A379599 - cp's OEIS frontend

A379598 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 A110447.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 6, 0, 1, 4, 9, 16, 23, 0, 1, 5, 14, 31, 62, 104, 0, 1, 6, 20, 52, 123, 278, 531, 0, 1, 7, 27, 80, 213, 552, 1398, 2982, 0, 1, 8, 35, 116, 340, 964, 2750, 7718, 18109, 0, 1, 9, 44, 161, 513, 1561, 4784, 14976, 46083, 117545, 0

Offset: 0

Seiichi Manyama, Feb 27 2025

			Square array begins:
  1,   1,    1,    1,    1,    1,     1, ...
  0,   1,    2,    3,    4,    5,     6, ...
  0,   2,    5,    9,   14,   20,    27, ...
  0,   6,   16,   31,   52,   80,   116, ...
  0,  23,   62,  123,  213,  340,   513, ...
  0, 104,  278,  552,  964, 1561,  2400, ...
  0, 531, 1398, 2750, 4784, 7755, 11987, ...

Columns k=0..1 give A000007, A110447 (A030266(n+1)).

Cf. A379599, A380178.

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

See A030266.

A381594 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 A381601.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 7, 0, 1, 3, 15, 79, 0, 1, 4, 24, 172, 1134, 0, 1, 5, 34, 280, 2475, 18953, 0, 1, 6, 45, 404, 4044, 41280, 353134, 0, 1, 7, 57, 545, 5863, 67365, 766291, 7154751, 0, 1, 8, 70, 704, 7955, 97620, 1246534, 15460284, 155181240, 0, 1, 9, 84, 882, 10344, 132486, 1801536, 25051422, 333896388, 3565276582, 0

Offset: 0

Seiichi Manyama, Mar 01 2025

			Square array begins:
  1,      1,      1,       1,       1,       1,       1, ...
  0,      1,      2,       3,       4,       5,       6, ...
  0,      7,     15,      24,      34,      45,      57, ...
  0,     79,    172,     280,     404,     545,     704, ...
  0,   1134,   2475,    4044,    5863,    7955,   10344, ...
  0,  18953,  41280,   67365,   97620,  132486,  172434, ...
  0, 353134, 766291, 1246534, 1801536, 2439615, 3169770, ...

Columns k=0..1 give A000007, A381601.

Cf. A379599, A381573, A381592.

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

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

A384623 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 A384622.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 7, 0, 1, 3, 15, 75, 0, 1, 4, 24, 164, 989, 0, 1, 5, 34, 268, 2177, 14822, 0, 1, 6, 45, 388, 3585, 32672, 242833, 0, 1, 7, 57, 525, 5235, 53922, 534781, 4253818, 0, 1, 8, 70, 680, 7150, 78972, 882304, 9349160, 78573475, 0, 1, 9, 84, 854, 9354, 108251, 1292456, 15399930, 172255669, 1516124048, 0

Offset: 0

Seiichi Manyama, Jun 05 2025

			Square array begins:
  1,      1,      1,      1,       1,       1,       1, ...
  0,      1,      2,      3,       4,       5,       6, ...
  0,      7,     15,     24,      34,      45,      57, ...
  0,     75,    164,    268,     388,     525,     680, ...
  0,    989,   2177,   3585,    5235,    7150,    9354, ...
  0,  14822,  32672,  53922,   78972,  108251,  142218, ...
  0, 242833, 534781, 882304, 1292456, 1772920, 2332044, ...

Columns k=0..1 give A000007, A384622.

Cf. A379599, A384619, A384620, A384621.

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

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

A381592 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 A381600.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, 39, 0, 1, 4, 18, 88, 383, 0, 1, 5, 26, 148, 869, 4360, 0, 1, 6, 35, 220, 1473, 9876, 55201, 0, 1, 7, 45, 305, 2211, 16740, 124473, 758877, 0, 1, 8, 56, 404, 3100, 25164, 210260, 1701630, 11157081, 0, 1, 9, 68, 518, 4158, 35381, 315312, 2860317, 24870695, 173623407, 0

Offset: 0

Seiichi Manyama, Mar 01 2025

			Square array begins:
  1,     1,      1,      1,      1,      1,      1, ...
  0,     1,      2,      3,      4,      5,      6, ...
  0,     5,     11,     18,     26,     35,     45, ...
  0,    39,     88,    148,    220,    305,    404, ...
  0,   383,    869,   1473,   2211,   3100,   4158, ...
  0,  4360,   9876,  16740,  25164,  35381,  47646, ...
  0, 55201, 124473, 210260, 315312, 442710, 595892, ...

Columns k=0..2 give A000007, A381600, A381593.

Cf. A379599, A381571, A381594, A381602.

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

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

A384619 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of (B(x)/x)^k, where B(x) is the g.f. of A213591.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 24, 0, 1, 4, 15, 56, 178, 0, 1, 5, 22, 97, 420, 1512, 0, 1, 6, 30, 148, 738, 3572, 14152, 0, 1, 7, 39, 210, 1145, 6300, 33328, 142705, 0, 1, 8, 49, 284, 1655, 9832, 58702, 334354, 1528212, 0, 1, 9, 60, 371, 2283, 14321, 91640, 586635, 3559310, 17211564, 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,     4,     9,    15,    22,     30,     39, ...
  0,    24,    56,    97,   148,    210,    284, ...
  0,   178,   420,   738,  1145,   1655,   2283, ...
  0,  1512,  3572,  6300,  9832,  14321,  19938, ...
  0, 14152, 33328, 58702, 91640, 133720, 186753, ...

Columns k=0..1 give A000007, A213591(n+1).

Cf. A379599, A384620, A384621, A384623.

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

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

A384620 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of (B(x)/x)^k, where B(x) is the g.f. of A213639.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, 38, 0, 1, 4, 18, 86, 357, 0, 1, 5, 26, 145, 815, 3832, 0, 1, 6, 35, 216, 1389, 8758, 45189, 0, 1, 7, 45, 300, 2095, 14967, 103056, 572378, 0, 1, 8, 56, 398, 2950, 22668, 175937, 1300586, 7676653, 0, 1, 9, 68, 511, 3972, 32091, 266470, 2214012, 17368633, 107971691, 0

Offset: 0

Seiichi Manyama, Jun 05 2025

			Square array begins:
  1,     1,      1,      1,      1,      1,      1, ...
  0,     1,      2,      3,      4,      5,      6, ...
  0,     5,     11,     18,     26,     35,     45, ...
  0,    38,     86,    145,    216,    300,    398, ...
  0,   357,    815,   1389,   2095,   2950,   3972, ...
  0,  3832,   8758,  14967,  22668,  32091,  43488, ...
  0, 45189, 103056, 175937, 266470, 377620, 512705, ...

Columns k=0..1 give A000007, A213639(n+1).

Cf. A379599, A384619, A384621, A384623.

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

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

A384621 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of (B(x)/x)^k, where B(x) is the g.f. of A376176.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 6, 0, 1, 3, 13, 55, 0, 1, 4, 21, 122, 622, 0, 1, 5, 30, 202, 1390, 8015, 0, 1, 6, 40, 296, 2322, 17934, 113164, 0, 1, 7, 51, 405, 3437, 30030, 252847, 1711898, 0, 1, 8, 63, 530, 4755, 44600, 423111, 3814724, 27357970, 0, 1, 9, 76, 672, 6297, 61966, 628454, 6369930, 60766238, 457507917, 0

Offset: 0

Seiichi Manyama, Jun 05 2025

			Square array begins:
  1,      1,      1,      1,      1,      1,       1, ...
  0,      1,      2,      3,      4,      5,       6, ...
  0,      6,     13,     21,     30,     40,      51, ...
  0,     55,    122,    202,    296,    405,     530, ...
  0,    622,   1390,   2322,   3437,   4755,    6297, ...
  0,   8015,  17934,  30030,  44600,  61966,   82476, ...
  0, 113164, 252847, 423111, 628454, 873840, 1164730, ...

Columns k=0..1 give A000007, A376176(n+1).

Cf. A379599, A384619, A384620, A384623.

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

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

cp's OEIS Frontend

A379598 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 A110447.

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A381594 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 A381601.

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A384623 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 A384622.

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A381592 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 A381600.

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A384619 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of (B(x)/x)^k, where B(x) is the g.f. of A213591.

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A384620 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of (B(x)/x)^k, where B(x) is the g.f. of A213639.

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A384621 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of (B(x)/x)^k, where B(x) is the g.f. of A376176.

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