A365673 -id:A365673 - cp's OEIS frontend

A366137 Main diagonal of A365673.

1, 1, 3, 34, 1385, 151416, 34988647, 14652451360, 10043767063521, 10488938037348736, 15823297170233169371, 33093543421485325857024, 92866643063435821361785225, 340452463858281665211769947136, 1594758692956372710911812262365695, 9367635873493885183992990317902544896

Offset: 0

Peter Luschny, Sep 30 2023

nonn

See the formulas in A365673.

Cf. A365673.

# Using function T from A365673.
seq(T(n, n, n), n = 0..15);

A366149 Triangle read by rows. T(n, k) = A000566(n - k + 1) * T(n, k - 1) + T(n - 1, k) for 0 < k < n. T(n, 0) = 1 and T(n, n) = T(n, n - 1) if n > 0.

Original entry on oeis.org

1, 1, 1, 1, 8, 8, 1, 26, 190, 190, 1, 60, 1270, 9080, 9080, 1, 115, 5180, 102320, 725320, 725320, 1, 196, 15960, 644960, 12334600, 87067520, 87067520, 1, 308, 40908, 2894900, 110761200, 2080769120, 14652451360, 14652451360

Offset: 0

Peter Luschny, Oct 01 2023

This a weighted generalized Catalan triangle (A365673) with the heptagonal numbers as weights.

			Triangle T(n, k) starts:
[0] 1;
[1] 1,   1;
[2] 1,   8,     8;
[3] 1,  26,   190,     190;
[4] 1,  60,  1270,    9080,      9080;
[5] 1, 115,  5180,  102320,    725320,     725320;
[6] 1, 196, 15960,  644960,  12334600,   87067520,    87067520;
[7] 1, 308, 40908, 2894900, 110761200, 2080769120, 14652451360, 14652451360;

Cf. A000566, A366150 (main diagonal), A365673 (general case).

T := proc(n, k) option remember;
if k = 0 then 1 else if k = n then T(n, k-1) else
(((5*k - 5*n - 2)*(k - n - 1))/2) * T(n, k - 1) + T(n - 1, k) fi fi end:
seq(seq(T(n, k), k = 0..n), n = 0..8);

A366149[n_, k_] := A366149[n, k] = Which[k==0, 1, k==n, A366149[n, k-1], True, PolygonalNumber[7, n-k+1] A366149[n, k-1] + A366149[n-1, k]];
Table[A366149[n, k], {n,0,10}, {k,0,n}] (* Paolo Xausa, Jan 01 2024 *)

A365672 Triangle read by rows. T(n, k) = 1 if k = 0, equals T(n, k-1) if k = n, and otherwise is (n - k + 1) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k).

Original entry on oeis.org

1, 1, 1, 1, 7, 7, 1, 22, 139, 139, 1, 50, 889, 5473, 5473, 1, 95, 3549, 58708, 357721, 357721, 1, 161, 10794, 360940, 5771821, 34988647, 34988647, 1, 252, 27426, 1595110, 50434901, 791512162, 4784061619, 4784061619

Offset: 0

Peter Luschny, Sep 29 2023

This triangle is described by Peter Bala (see link).

This a weighted generalized Catalan triangle (A365673) with the hexagonal numbers as weights.

			Triangle T(n, k) starts:
[0] 1;
[1] 1,   1;
[2] 1,   7,     7;
[3] 1,  22,   139,     139;
[4] 1,  50,   889,    5473,     5473;
[5] 1,  95,  3549,   58708,   357721,    357721;
[6] 1, 161, 10794,  360940,  5771821,  34988647,   34988647;
[7] 1, 252, 27426, 1595110, 50434901, 791512162, 4784061619, 4784061619;

Peter Bala, A triangle for calculating A126156.

Cf. A000384, A126156 (main diagonal), A365673 (general case).

T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1) else (n - k + 1) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k) fi fi end:

A366150 a(n) = A366149(n, n).

Original entry on oeis.org

1, 1, 8, 190, 9080, 725320, 87067520, 14652451360, 3291452374400, 951504315644800, 344097665398707200, 152191547632887731200, 80821740082953347993600, 50749489515964112842240000, 37193794460546391363092480000, 31464706192542487393274091520000

Offset: 0

Peter Luschny, Oct 01 2023

nonn

This a weighted generalized Catalan sequence (A365673) with the heptagonal numbers as weights.

Cf. A366149, A365673.

# Using function T from A366149.
a := n -> T(n, n): seq(a(n), n = 0..16);

A366149[n_,k_]:=A366149[n,k]=Which[k==0,1,k==n,A366149[n,k-1],True,PolygonalNumber[7,n-k+1]A366149[n,k-1]+A366149[n-1,k]];
Array[A366149[#,#]&,20,0] (* Paolo Xausa, Jan 01 2024 *)

A365674 Triangle read by rows. T(n, k) = ((n - k + 1)(n - k + 2)/2) T(n, k - 1) + T(n - 1, k) for 0 < k < n, T(n, 0) = 1 and T(n, n) = T(n, n - 1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 10, 34, 34, 1, 20, 154, 496, 496, 1, 35, 504, 3520, 11056, 11056, 1, 56, 1344, 16960, 112816, 349504, 349504, 1, 84, 3108, 63580, 748616, 4841200, 14873104, 14873104, 1, 120, 6468, 199408, 3739736, 42238560, 268304464, 819786496, 819786496

Offset: 0

Peter Luschny, Sep 30 2023

This triangle is associated to the case n = 3 of A365673 and has as weight function the triangular numbers A000217. The numbers on its main diagonal are the reduced tangent numbers A002105. For details see A365673.

			[0] 1;
[1] 1,  1;
[2] 1,  4,    4;
[3] 1, 10,   34,    34;
[4] 1, 20,  154,   496,    496;
[5] 1, 35,  504,  3520,  11056,   11056;
[6] 1, 56, 1344, 16960, 112816,  349504,   349504;
[7] 1, 84, 3108, 63580, 748616, 4841200, 14873104, 14873104;

Cf. A002105 (main diagonal), A365673 (case n=3), A000217 (weight).

T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k - 1) else ((n - k + 1)*(n - k + 2)/2) * T( n, k - 1) + T( n - 1, k) fi fi end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);

A366138 Triangle read by rows. T(n, k) = A000326(n - k + 1) * T(n, k - 1) + T(n - 1, k) for 0 < k < n. T(n, 0) = 1 and T(n, n) = T(n, n - 1) if n > 0.

Original entry on oeis.org

1, 1, 1, 1, 6, 6, 1, 18, 96, 96, 1, 40, 576, 2976, 2976, 1, 75, 2226, 29688, 151416, 151416, 1, 126, 6636, 175680, 2259576, 11449296, 11449296, 1, 196, 16632, 757800, 18931176, 238623408, 1204566336, 1204566336

Offset: 0

Peter Luschny, Oct 01 2023

This a weighted generalized Catalan triangle (A365673) with the pentagonal numbers as weights.

			Triangle T(n, k) starts:
[0] 1;
[1] 1,   1;
[2] 1,   6,     6;
[3] 1,  18,    96,     96;
[4] 1,  40,   576,   2976,     2976;
[5] 1,  75,  2226,  29688,   151416,    151416;
[6] 1, 126,  6636, 175680,  2259576,  11449296,   11449296;
[7] 1, 196, 16632, 757800, 18931176, 238623408, 1204566336, 1204566336;

Cf. A000326, A126151 (main diagonal), A365673.

T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (((n - k + 1)*(3*n - 3*k + 2))/2) * T(n, k - 1) + T(n - 1, k) fi fi end:
seq(seq(T(n, k), k = 0..n), n = 0..8);

cp's OEIS Frontend

A366137 Main diagonal of A365673.

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A366149 Triangle read by rows. T(n, k) = A000566(n - k + 1) * T(n, k - 1) + T(n - 1, k) for 0 < k < n. T(n, 0) = 1 and T(n, n) = T(n, n - 1) if n > 0.

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A365672 Triangle read by rows. T(n, k) = 1 if k = 0, equals T(n, k-1) if k = n, and otherwise is (n - k + 1) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k).

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A366150 a(n) = A366149(n, n).

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Mathematica

A365674 Triangle read by rows. T(n, k) = ((n - k + 1)*(n - k + 2)/2) * T(n, k - 1) + T(n - 1, k) for 0 < k < n, T(n, 0) = 1 and T(n, n) = T(n, n - 1) for n > 0.

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Maple

A366138 Triangle read by rows. T(n, k) = A000326(n - k + 1) * T(n, k - 1) + T(n - 1, k) for 0 < k < n. T(n, 0) = 1 and T(n, n) = T(n, n - 1) if n > 0.

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Maple

A365674 Triangle read by rows. T(n, k) = ((n - k + 1)(n - k + 2)/2) T(n, k - 1) + T(n - 1, k) for 0 < k < n, T(n, 0) = 1 and T(n, n) = T(n, n - 1) for n > 0.