A293985 -id:A293985 - cp's OEIS frontend

A361616 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*(j+1),n-j)/j!.

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 15, 34, 1, 1, 5, 25, 103, 209, 1, 1, 6, 37, 214, 885, 1546, 1, 1, 7, 51, 373, 2293, 9051, 13327, 1, 1, 8, 67, 586, 4721, 29176, 106843, 130922, 1, 1, 9, 85, 859, 8481, 70981, 427189, 1425495, 1441729, 1

Offset: 0

Seiichi Manyama, Mar 18 2023

			Square array begins:
  1,    1,    1,     1,     1,      1, ...
  1,    2,    3,     4,     5,      6, ...
  1,    7,   15,    25,    37,     51, ...
  1,   34,  103,   214,   373,    586, ...
  1,  209,  885,  2293,  4721,   8481, ...
  1, 1546 ,9051, 29176, 70981, 146046, ...

Columns k=0..3 give A000012, A002720, A343884, A351767.
Main diagonal gives A361617.
Cf. A293985, A361600.

T(n,k) = n! * sum(j=0, n, binomial(n+(k-1)*(j+1), n-j)/j!);

E.g.f. of column k: exp( x/(1-x)^k ) / (1-x)^k.

T(n,k) = Sum_{j=0..n} (n+(k-1)*(j+1))!/(k*j+k-1)! * binomial(n,j) for k > 0.

A380491 a(n) = n! * Sum_{k=0..n} binomial(2*n-3,k)/(n-k)!.

Original entry on oeis.org

1, 0, 3, 34, 501, 9276, 207775, 5470158, 165625929, 5671386136, 216730118331, 9144481575450, 422249317829053, 21180324426577044, 1146880568461500951, 66677192513929212166, 4142571510546929867025, 273910161452560881843888, 19204878684852222745880179

Offset: 0

Seiichi Manyama, Jan 25 2025

nonn

Cf. A082545, A152059, A251568, A293985, A343832, A380492, A380493.

PARI
```
a(n) = n!*pollaguerre(n, n-3, -1);
```

a(n) = n! * LaguerreL(n, n-3, -1).

a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n-2).

A380492 a(n) = n! * Sum_{k=0..n} binomial(2*n-2,k)/(n-k)!.

Original entry on oeis.org

1, 1, 7, 73, 1045, 19081, 424051, 11109337, 335262313, 11453449105, 436944953791, 18412283563081, 849345673881277, 42570185481576793, 2303643608370636715, 133859418832759525081, 8312945340897388101841, 549460711493172343519777, 38513032385247860120975863

Offset: 0

Seiichi Manyama, Jan 25 2025

nonn

Cf. A082545, A152059, A251568, A293985, A343832, A380491, A380493.

PARI
```
a(n) = n!*pollaguerre(n, n-2, -1);
```

a(n) = n! * LaguerreL(n, n-2, -1).

a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n-1).

A380493 a(n) = n! * Sum_{k=0..n} binomial(2*n+3,k)/(n-k)!.

Original entry on oeis.org

1, 6, 57, 748, 12585, 259026, 6315001, 178134552, 5711078673, 205209960670, 8171229107481, 357235056697476, 17014791129640057, 877089297426429738, 48657292133825026905, 2890717184573264397616, 183125115830192864360481, 12323226433255671469949622

Offset: 0

Seiichi Manyama, Jan 25 2025

nonn

Cf. A082545, A152059, A251568, A293985, A343832, A380491, A380492.

PARI
```
a(n) = n!*pollaguerre(n, n+3, -1);
```

a(n) = n! * LaguerreL(n, n+3, -1).

a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n+4).

cp's OEIS Frontend

A361616 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*(j+1),n-j)/j!.

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PARI

Formula

A380491 a(n) = n! * Sum_{k=0..n} binomial(2*n-3,k)/(n-k)!.

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Author

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Links

Crossrefs

Programs

PARI

Formula

A380492 a(n) = n! * Sum_{k=0..n} binomial(2*n-2,k)/(n-k)!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A380493 a(n) = n! * Sum_{k=0..n} binomial(2*n+3,k)/(n-k)!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula