A058006 -id:A058006 - cp's OEIS frontend

A176663 T(n, k) = [x^k] Sum_{j=0..n} j!*binomial(x, j), for 0 <= k <= n, triangle read by rows.

1, 1, 1, 1, 0, 1, 1, 2, -2, 1, 1, -4, 9, -5, 1, 1, 20, -41, 30, -9, 1, 1, -100, 233, -195, 76, -14, 1, 1, 620, -1531, 1429, -659, 161, -20, 1, 1, -4420, 11537, -11703, 6110, -1799, 302, -27, 1, 1, 35900, -98047, 106421, -61174, 20650, -4234, 519, -35, 1

Offset: 0

Roger L. Bagula, Apr 23 2010

			Triangle starts:
  {1},
  {1,       1},
  {1,       0,      1},
  {1,       2,     -2,        1},
  {1,      -4,      9,       -5,      1},
  {1,      20,    -41,       30,     -9,       1},
  {1,    -100,    233,     -195,     76,     -14,     1},
  {1,     620,  -1531,     1429,   -659,     161,   -20,     1},
  {1,   -4420,  11537,   -11703,   6110,   -1799,   302,   -27,   1},
  {1,   35900, -98047,   106421, -61174,   20650, -4234,   519, -35,   1},
  {1, -326980, 928529, -1066279, 662506, -248675, 59039, -8931, 835, -44, 1}

Row sums are A040000. Alternating row sums are A058006, which are also T(n,1).

Cf. A186020.

with(PolynomialTools):
T_row := n -> CoefficientList(expand(add(k!*binomial(x, k), k=0..n)), x):
ListTools:-Flatten([seq(T_row(n), n=0..9)]); # Peter Luschny, Jul 02 2019

p[x_, n_] := Sum[k! Binomial[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[x, n]], x], {n, 0, 10}] // Flatten
(* Alternative: *)
Table[CoefficientList[FunctionExpand[Sum[FactorialPower[x, k], {k, 0, n}]], x], {n, 0, 10}] // Flatten (* Peter Luschny, Jul 02 2019 *)

From Peter Luschny, Jul 02 2019: (Start)
Sum_{k=0..n} T(n, k)*x^k = Sum_{k=0..n} (x)_k, where (x)_k denotes the falling factorial.
Let T be the lower triangular matrix associated to the T(n, k) and S the lower triangular matrix associated to the Stirling set numbers S2(n, k). Then S*T = A186020 (seen as a matrix) and T*S = A000012 (seen as a matrix). (End)
T(n, k) = Sum_{i=0..n-k} Stirling1(i+k, k). - Igor Victorovich Statsenko, May 25 2024

Edited by Peter Luschny, Jul 02 2019

A260612 Triangle read by rows: T(n,k) = coefficient of x^(n-k) in Product_{m=0..n-1} (x+(-1)^m*m!), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 0, -1, 1, 2, -1, -2, 1, -4, -13, 4, 12, 1, 20, -109, -308, 108, 288, 1, -100, -2509, 12772, 37068, -12672, -34560, 1, 620, -74509, -1793708, 9232908, 26676288, -9158400, -24883200, 1, -4420, -3199309, 373731652, 9049521228, -46507180032, -134457649920, 46133452800, 125411328000

Offset: 0

Matthew Campbell, Aug 08 2015

			Row 0: 1.
Row 1: (x+(-1)^(0)*0!) = x+1. Coefficients are 1 and 1.
Row 2: (x+(-1)^0*0!)*(x+(-1)^(1)*1!) = (x+1)*(x-1) = x^2-1. Coefficients are 1, 0, and -1.
Row 3: (x+(-1)^(0)*0!)*(x-(-1)^(1)*1!)*(x+(-1)^(2)*2!) = (x+1)*(x-1)*(x+2) = x^3 + 2*x^2 - x - 2. Coefficients are 1, 2, -1, and -2.

Jon E. Schoenfield, Table of n, a(n) for n = 0..1034 (first 120 terms from Matthew Campbell)

Cf. A058006, A153229.

T(n, 1) = A058006(n-1) = (-1)^(n+1)*A153229(n) for n >= 1.

A368555 a(n) = Sum_{k=0..n} k! * 3^(n-k).

Original entry on oeis.org

1, 4, 14, 48, 168, 624, 2592, 12816, 78768, 599184, 5426352, 56195856, 647589168, 8169788304, 111687656112, 1642737336336, 25851001897008, 433240433787024, 7702095007089072, 144751385430099216, 2867156164466937648, 59692410665110252944

Offset: 0

Seiichi Manyama, Dec 29 2023

Cf. A003422, A058006, A112370, A112541, A233449.

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

a(0) = 1; a(n) = 3*a(n-1) + n!.
a(n) = (n+3)*a(n-1) - 3*n*a(n-2).
a(n) ~ n!. - Vaclav Kotesovec, Jan 13 2024

A064493 List of distinct greatest common divisors of { k! and k! - (k-1)! + (k-2)! - ... + (-1)^k*0!, k = 0 .. n}.

Original entry on oeis.org

1, 2, 4, 20, 260, 9620, 4454060

Offset: 0

Vladeta Jovovic, Oct 04 2001

Records in the sequence gcd(k!,A058006(k)). The record indices are products of primes from A064384. There are no further terms unless there are terms in A064384 larger than 463. - Max Alekseyev, Jul 06 2010

Cf. A000142, A058006, A064383, A064384, A005165.

A368573 a(n) = Sum_{k=0..n} k! * k^k.

Original entry on oeis.org

1, 2, 10, 172, 6316, 381316, 33973636, 4184630356, 680641979476, 141267789027796, 36429267789027796, 11425158161234192596, 4282251538636575402196, 1890291840090813125264596, 970616058694975134467523796, 573593232413546537634467523796, 386530943604347679737893326531796

Offset: 0

Seiichi Manyama, Dec 31 2023

Partial sums of A061711.
Cf. A058006, A112368, A112370.
Cf. A367011, A367012, A368561.

PARI
```
a(n) = sum(k=0, n, k!*k^k);
```

a(n) ~ n! * n^n. - Vaclav Kotesovec, Jan 13 2024

cp's OEIS Frontend

A176663 T(n, k) = [x^k] Sum_{j=0..n} j!*binomial(x, j), for 0 <= k <= n, triangle read by rows.

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A260612 Triangle read by rows: T(n,k) = coefficient of x^(n-k) in Product_{m=0..n-1} (x+(-1)^m*m!), 0 <= k <= n.

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A368555 a(n) = Sum_{k=0..n} k! * 3^(n-k).

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A064493 List of distinct greatest common divisors of { k! and k! - (k-1)! + (k-2)! - ... + (-1)^k*0!, k = 0 .. n}.

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A368573 a(n) = Sum_{k=0..n} k! * k^k.

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