A122852 -id:A122852 - cp's OEIS frontend

A072374 a(1) = 1; a(n) = 1 + Sum_{i=1..n} Product_{j=i..2*i-1} (n-j).

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: 1

N. J. A. Sloane, Jul 19 2002

nonn

A122852 is another version of the same sequence. - R. J. Mathar, Jun 18 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Table[Sum[Binomial[n-k,k]*k!,{k,0,Floor[n/2]}],{n,1,20}] (* Vaclav Kotesovec, Feb 08 2014 *)

The sequence 1, 1, 2, 3, .. has a(n)=sum{k=0..floor(n/2), C(n-k, k)k!} (diagonal sums of permutation triangle A008279). - Paul Barry, May 12 2004
Recurrence: 2*a(n) = 3*a(n-1) + (n-1)*a(n-2) - (n-1)*a(n-3). - Vaclav Kotesovec, Feb 08 2014
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

A357532 a(n) = Sum_{k=0..floor(n/3)} (n-2k)!/(n-3k)!.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 12, 19, 34, 63, 112, 211, 414, 799, 1588, 3267, 6706, 13999, 30024, 64723, 141142, 314271, 705724, 1599619, 3685338, 8573167, 20112016, 47804499, 114743614, 277615903, 679057092, 1676636611, 4171532674, 10477002159, 26545428568, 67755344467, 174386589606

Offset: 0

Seiichi Manyama, Nov 19 2022

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

Cf. A072374, A122852, A357533, A357570.

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

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

a(n) ~ c * n^(n/3 + 1/2) / (3^(n/3) * exp(n/3 - n^(2/3)/3^(2/3) - 2*n^(1/3) / 3^(7/3))) * (1 + 1235/(729 * 3^(2/3) * n^(1/3)) + 9452027/(15943230 * 3^(1/3) * n^(2/3)) + 16015315669/(41841412812*n)), where c = 0.50682110703119..., conjecture: c = exp(4/81) * sqrt(2*Pi) / 3^(3/2). - Vaclav Kotesovec, Nov 25 2022

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.

A122851 Number triangle T(n,k) = C(k,n-k)*(n-k)!.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 6, 4, 1, 0, 0, 0, 6, 12, 5, 1, 0, 0, 0, 0, 24, 20, 6, 1, 0, 0, 0, 0, 24, 60, 30, 7, 1, 0, 0, 0, 0, 0, 120, 120, 42, 8, 1, 0, 0, 0, 0, 0, 120, 360, 210, 56, 9, 1, 0, 0, 0, 0, 0, 0, 720, 840, 336, 72, 10, 1

Offset: 0

Paul Barry, Sep 14 2006

Row sums are A122852.

Triangle T(n,k), read by rows, given by (0,1,-1,0,0,1,-1,0,0,1,-1,0,0,1,...) DELTA (1,0,0,-1,2,0,0,-2,3,0,0,-3,4,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 0, 2, 1;
  0, 0, 2, 3,  1;
  0, 0, 0, 6,  4,  1;
  0, 0, 0, 6, 12,  5, 1;
  0, 0, 0, 0, 24, 20, 6, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A094587, A242653.

T(2n,n) gives A000142.

/* As triangle: */ [[Binomial(k,n-k)*Factorial(n-k): k in [0..n]]: n in [0.. 7]]; // Vincenzo Librandi, Apr 24 2015

Flatten[Table[Binomial[k,n-k](n-k)!,{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 16 2012 *)

Number triangle T(n,k) = [k<=n]*k!/(2k-n)!.

T(n,k) = A008279(k,n-k). - Danny Rorabaugh, Apr 23 2015

A358603 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * (n-k)!/(n-2*k)!.

Original entry on oeis.org

1, 1, 0, -1, 0, 3, 2, -9, -12, 35, 78, -153, -544, 723, 4170, -3337, -35028, 10851, 320678, 57255, -3178152, -2190253, 33864546, 42120183, -385314460, -719159517, 4649508222, 12033407591, -59076411312, -204022615725, 784134861818, 3554417974647, -10768948801764

Offset: 0

Seiichi Manyama, Nov 23 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..899

Cf. A358604, A358605, A358606.

Cf. A122852.

nxt[{n_,a_,b_}]:={n+1,b,(b-a(n+1)+1)/2}; NestList[nxt,{1,1,1},40][[;;,2]] (* Harvey P. Dale, Jul 25 2024 *)

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

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

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

A344391 T(n, k) = binomial(n - k, k) * factorial(k), for n >= 0 and 0 <= k <= floor(n/2). Triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 6, 1, 5, 12, 6, 1, 6, 20, 24, 1, 7, 30, 60, 24, 1, 8, 42, 120, 120, 1, 9, 56, 210, 360, 120, 1, 10, 72, 336, 840, 720, 1, 11, 90, 504, 1680, 2520, 720, 1, 12, 110, 720, 3024, 6720, 5040, 1, 13, 132, 990, 5040, 15120, 20160, 5040

Offset: 0

Peter Luschny, May 17 2021

The antidiagonal representation of the falling factorials (A008279).

			[ 0] [1]
[ 1] [1]
[ 2] [1,  1]
[ 3] [1,  2]
[ 4] [1,  3,  2]
[ 5] [1,  4,  6]
[ 6] [1,  5, 12,   6]
[ 7] [1,  6, 20,  24]
[ 8] [1,  7, 30,  60,  24]
[ 9] [1,  8, 42, 120, 120]
[10] [1,  9, 56, 210, 360, 120]
[11] [1, 10, 72, 336, 840, 720]

Cf. A122852 (row sums).

Cf. A008279, A122851.

T := (n, k) -> pochhammer(n + 1 - 2*k, k):
seq(print(seq(T(n, k), k=0..n/2)), n = 0..11);

def T(n, k): return rising_factorial(n + 1 - 2*k, k)
def T(n, k): return (-1)^k*falling_factorial(2*k - n - 1, k)
def T(n, k): return binomial(n - k, k) * factorial(k)
print(flatten([[T(n, k) for k in (0..n//2)] for n in (0..11)]))

T(n, k) = RisingFactorial(n + 1 - 2*k, k).

T(n, k) = (-1)^k*FallingFactorial(2*k - n - 1, k).

cp's OEIS Frontend

A072374 a(1) = 1; a(n) = 1 + Sum_{i=1..n} Product_{j=i..2*i-1} (n-j).

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A357532 a(n) = Sum_{k=0..floor(n/3)} (n-2*k)!/(n-3*k)!.

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

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

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A122851 Number triangle T(n,k) = C(k,n-k)*(n-k)!.

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Magma

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A358603 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * (n-k)!/(n-2*k)!.

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

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A344391 T(n, k) = binomial(n - k, k) * factorial(k), for n >= 0 and 0 <= k <= floor(n/2). Triangle read by rows.

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Formula

A357532 a(n) = Sum_{k=0..floor(n/3)} (n-2k)!/(n-3k)!.

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)!.