A357532 -id:A357532 - cp's OEIS frontend

A122852 Row sums of number triangle A122851.

1, 1, 2, 3, 6, 11, 24, 51, 122, 291, 756, 1979, 5526, 15627, 46496, 140451, 442194, 1414931, 4687212, 15785451, 54764846, 193129659, 698978136, 2570480147, 9672977706, 36967490691, 144232455524, 571177352091, 2304843053382, 9434493132011, 39289892366736

Offset: 0

Paul Barry, Sep 14 2006

Essentially the same as A072374. - R. J. Mathar, Jun 18 2008

Diagonal sums of A008279. - Paul Barry, Feb 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jonathan Fang, Zachary Hamaker, and Justin Troyka, On pattern avoidance in matchings and involutions, arXiv:2009.00079 [math.CO], 2020. See Theorem 1.6 (b).
Guo-Niu Han, Hankel Continued fractions and Hankel determinants of the Euler numbers, arXiv:1906.00103 [math.CO], 2019. See p. 27.
Qiong Qiong Pan and Jiang Zeng, The gamma-coefficients of Branden's (p,q)-Eulerian polynomials and André permutations, arXiv:1910.01747 [math.CO], 2019.

Cf. A072374, A008279.

Cf. A357532, A357533, A357570.

Table[Sum[Binomial[n-k,k]*k!,{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Feb 08 2014 *)

a(n) = sum(k=0, n, binomial(k,n-k)*(n-k)!); \\ Michel Marcus, Sep 02 2020

a(n) = Sum{k=0..n} C(k,n-k)*(n-k)!.
From Paul Barry, Feb 11 2009: (Start)
G.f.: 1/(1-x-x^2/(1-x^2/(1-x-2x^2/(1-2x^2/(1-x-3x^2/(1-3x^2/(1-x-4x^2/(1-4x^2/(1-... (continued fraction).
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*k!. (End)
D-finite with recurrence -2*a(n) + 3*a(n-1) + (n-1)*a(n-2) + (-n+1)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012. Proof in [Han 2019]
a(n) ~ sqrt(Pi) * exp(sqrt(n/2) - n/2 + 1/8) * n^((n+1)/2) / 2^(n/2+1) * (1 + 37/(48*sqrt(2*n))). - Vaclav Kotesovec, Feb 08 2014
a(n) = (a(n-1) + n * a(n-2) + 1)/2 for n > 1. - Seiichi Manyama, Nov 19 2022

More terms from Vaclav Kotesovec, Jun 04 2019

A357533 a(n) = Sum_{k=0..floor(n/4)} (n-3k)!/(n-4k)!.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 20, 29, 46, 77, 128, 205, 338, 581, 1012, 1733, 2990, 5293, 9536, 17117, 30778, 56165, 104108, 193621, 360662, 677693, 1289080, 2467373, 4735826, 9142837, 17814308, 34950245, 68835118, 136197581, 271384112, 544302973, 1096578410, 2218459013, 4513377436

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A072374, A122852, A357532, A357570.

a(n) = sum(k=0, n\4, (n-3*k)!/(n-4*k)!);

a(n) = (3 * a(n-1) + n * a(n-4) + 1)/4 for n > 3.

A357570 a(n) = Sum_{k=0..floor(n/5)} (n-4k)!/(n-5k)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 14, 21, 30, 41, 60, 93, 146, 225, 336, 509, 798, 1281, 2060, 3261, 5154, 8273, 13536, 22365, 36806, 60369, 99588, 166301, 280650, 474801, 802424, 1358973, 2317806, 3987185, 6893196, 11933949, 20690738, 36022161, 63107520, 111146141, 196322454, 347412753

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A072374, A122852, A357532, A357533.

a(n) = sum(k=0, n\5, (n-4*k)!/(n-5*k)!);

D-finite with recurrence a(n) = (4 * a(n-1) + n * a(n-5) + 1)/5 for n > 4.

A358604 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (n-2k)!/(n-3k)!.

Original entry on oeis.org

1, 1, 1, 0, -1, -2, -1, 2, 7, 8, -1, -26, -49, -28, 103, 314, 359, -344, -2113, -3682, -161, 14684, 36791, 25762, -100297, -373456, -472241, 587846, 3877487, 7149988, -1111801, -40808566, -103472249, -56751688, 424662623, 1490284654, 1674543359, -4121143444

Offset: 0

Seiichi Manyama, Nov 23 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A358603, A358605, A358606.

Cf. A357532.

a(n) = sum(k=0, n\3, (-1)^k*(n-2*k)!/(n-3*k)!);

a(n) = (2 * a(n-1) - n * a(n-3) + 1)/3 for n > 2.

A358547 a(n) = Sum_{k=0..floor(n/3)} (n-k)!/(n-3*k)!.

Original entry on oeis.org

1, 1, 1, 3, 7, 13, 45, 151, 403, 1617, 6793, 23275, 105951, 522133, 2159077, 10964223, 61134955, 293587801, 1641566913, 10124731987, 55014334903, 335177088285, 2251814156701, 13587321392743, 89436553249347, 647267633012833, 4276528756374265, 30198747030078651

Offset: 0

Seiichi Manyama, Nov 21 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..673

Cf. A122852, A330044, A357532.

PARI
```
a(n) = sum(k=0, n\3, (n-k)!/(n-3*k)!);
```

a(n) = (3 * (2*n-1) * a(n-1) - n * a(n-2) + 2 * (n-1) * n * (2*n-3) * a(n-3) + 2 * (2*n-3))/(9 * (n-1)) for n > 2.

a(n) ~ sqrt(Pi) * 2^(2*n/3 + 1) * n^(2*n/3 + 1/2) / (3^(2*n/3 + 3/2) * exp(2*n/3 - (2/3)^(1/3) * n^(1/3))) * (1 + 1/(2^(4/3) * 3^(5/3) * n^(1/3)) + 145/(2^(11/3) * 3^(10/3) * n^(2/3)) + 3349/(23328*n)). - Vaclav Kotesovec, Nov 25 2022

cp's OEIS Frontend

A122852 Row sums of number triangle A122851.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A357533 a(n) = Sum_{k=0..floor(n/4)} (n-3*k)!/(n-4*k)!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A357570 a(n) = Sum_{k=0..floor(n/5)} (n-4*k)!/(n-5*k)!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A358604 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (n-2*k)!/(n-3*k)!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A358547 a(n) = Sum_{k=0..floor(n/3)} (n-k)!/(n-3*k)!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A357533 a(n) = Sum_{k=0..floor(n/4)} (n-3k)!/(n-4k)!.

A357570 a(n) = Sum_{k=0..floor(n/5)} (n-4k)!/(n-5k)!.

A358604 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (n-2k)!/(n-3k)!.