A060058 -id:A060058 - cp's OEIS frontend

A371996 Triangle read by rows: T(m, n, k) = 1 if k = 0 and T(m, n, k - 1) if k = n; otherwise (-1)^m(k - n - 1)^m T(m, n, k - 1) + T(m, n - 1, k) where m = 3.

1, 1, 1, 1, 9, 9, 1, 36, 297, 297, 1, 100, 2997, 24273, 24273, 1, 225, 17397, 493992, 3976209, 3976209, 1, 441, 72522, 5135400, 142632009, 1145032281, 1145032281, 1, 784, 241866, 35368650, 2406225609, 66113123724, 530050022073, 530050022073

Offset: 0

Peter Luschny, Apr 24 2024

			Triangle read by rows:
[0] 1;
[1] 1,   1;
[2] 1,   9,     9;
[3] 1,  36,   297,     297;
[4] 1, 100,  2997,   24273,     24273;
[5] 1, 225, 17397,  493992,   3976209,    3976209;
[6] 1, 441, 72522, 5135400, 142632009, 1145032281, 1145032281;

Family: A009766 (Catalan's triangle, m=0), A001498 (m=1), A060058 (m=2), this triangle (m=3), A371997 (m=4).

Cf. A216966 (main diagonal).

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

A371997 Triangle read by rows: T(m, n, k) = 1 if k = 0 and T(m, n, k - 1) if k = n; otherwise (-1)^m(k - n - 1)^m T(m, n, k - 1) + T(m, n - 1, k) where m = 4.

Original entry on oeis.org

1, 1, 1, 1, 17, 17, 1, 98, 1585, 1585, 1, 354, 30259, 485729, 485729, 1, 979, 280883, 23237252, 372281761, 372281761, 1, 2275, 1702758, 459143300, 37562889061, 601378506737, 601378506737, 1, 4676, 7762854, 5310927050, 1397160213861, 113771355829478, 1820943071778385, 1820943071778385

Offset: 0

Peter Luschny, Apr 24 2024

			Triangle begins:
[0] 1;
[1] 1,    1;
[2] 1,   17,      17;
[3] 1,   98,    1585,      1585;
[4] 1,  354,   30259,    485729,      485729;
[5] 1,  979,  280883,  23237252,   372281761,    372281761;
[6] 1, 2275, 1702758, 459143300, 37562889061, 601378506737, 601378506737;

Family: A009766 (Catalan's triangle, m=0), A001498 (m=1), A060058 (m=2), A371996 (m=3), this triangle (m=4).

Cf. A227887 (main diagonal).

T := proc(m, n, k) option remember; if k = 0 then 1 elif k = n then T(m, n, k-1) else (-1)^m*(k - n - 1)^m * T(m, n, k - 1) + T(m, n - 1, k) fi end:
seq(seq(T(4, n, k), k = 0..n), n = 0..8));

A060076 Fourth column of triangle A060074.

Original entry on oeis.org

1, 30, 1211, 68060, 5162421, 510964090, 64108947631, 9954077496120, 1874917179220841, 421327494210872150, 111374995513203846051, 34217341079365673500180, 12091183195611248756287261

Offset: 0

Wolfdieter Lang, Mar 16 2001

Also fourth diagonal of triangle A060058.

a(n) is n-fold iterated sum of squares with last upper summation index 4 and other upper summation indices j[k]+1, k=2,...,n. See Comments A060061 for the threefold iteration with last upper summation index n.

a(n)=A060074(n+3, 3)= A060058(3+n, n).

A060077 Fifth column of triangle A060074.

Original entry on oeis.org

1, 55, 3486, 281210, 28862471, 3706931865, 584856590956, 111432850130020, 25255179275602941, 6719606199432053675, 2074965724098955112426, 736196253336729035680830, 297480985088238886530733411

Offset: 0

Wolfdieter Lang, Mar 16 2001

Also fifth diagonal of triangle A060058.

a(n) is n-fold iterated sum of squares with last upper summation index 5 and other upper summation indices j[k]+1, k=2,...,n. See Comments A060061 for the threefold iteration with last upper summation index n.

Cf. A060058, A060061, A060074.

a(n) = A060074(n+4, 4) = A060058(4+n, n).

A060078 Sixth column of triangle A060074.

Original entry on oeis.org

1, 91, 8526, 948002, 127838711, 20829905733, 4059150905356, 935210483855284, 251931148208847261, 78543629143430050655, 28079156490294534487226, 11415592075705105510977846, 5238681589930616835789832051

Offset: 0

Wolfdieter Lang, Mar 16 2001

nonn

Also sixth diagonal of triangle A060058.

a(n) is n-fold iterated sum of squares with last upper summation index 6 and other upper summation indices j[k]+1, k=2,...,n. See Comments A060061 for the threefold iteration with last upper summation index n.

cp's OEIS Frontend

A371996 Triangle read by rows: T(m, n, k) = 1 if k = 0 and T(m, n, k - 1) if k = n; otherwise (-1)^m(k - n - 1)^m T(m, n, k - 1) + T(m, n - 1, k) where m = 3.

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A371997 Triangle read by rows: T(m, n, k) = 1 if k = 0 and T(m, n, k - 1) if k = n; otherwise (-1)^m(k - n - 1)^m T(m, n, k - 1) + T(m, n - 1, k) where m = 4.

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A060076 Fourth column of triangle A060074.

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A060077 Fifth column of triangle A060074.

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A060078 Sixth column of triangle A060074.

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cp's OEIS Frontend

A371996 Triangle read by rows: T(m, n, k) = 1 if k = 0 and T(m, n, k - 1) if k = n; otherwise (-1)^m*(k - n - 1)^m * T(m, n, k - 1) + T(m, n - 1, k) where m = 3.

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Author

Keywords

Examples

Crossrefs

Programs

Maple

A371997 Triangle read by rows: T(m, n, k) = 1 if k = 0 and T(m, n, k - 1) if k = n; otherwise (-1)^m*(k - n - 1)^m * T(m, n, k - 1) + T(m, n - 1, k) where m = 4.

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Author

Keywords

Examples

Crossrefs

Programs

Maple

A060076 Fourth column of triangle A060074.

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Author

Keywords

Comments

Formula

A060077 Fifth column of triangle A060074.

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Author

Keywords

Comments

Crossrefs

Formula

A060078 Sixth column of triangle A060074.

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Author

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Comments

A371996 Triangle read by rows: T(m, n, k) = 1 if k = 0 and T(m, n, k - 1) if k = n; otherwise (-1)^m(k - n - 1)^m T(m, n, k - 1) + T(m, n - 1, k) where m = 3.

A371997 Triangle read by rows: T(m, n, k) = 1 if k = 0 and T(m, n, k - 1) if k = n; otherwise (-1)^m(k - n - 1)^m T(m, n, k - 1) + T(m, n - 1, k) where m = 4.