A001048 -id:A001048 - cp's OEIS frontend

A276588 Square array A(row,col) = Sum_{k=0..row} binomial(row,k)*(1+col+k)!, read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

1, 2, 3, 6, 8, 11, 24, 30, 38, 49, 120, 144, 174, 212, 261, 720, 840, 984, 1158, 1370, 1631, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 362880, 403200, 448560, 499680, 557400, 622704, 696750, 780908, 876809, 3628800, 3991680, 4394880, 4843440, 5343120, 5900520, 6523224, 7219974, 8000882, 8877691

Offset: 0

Antti Karttunen, Sep 19 2016

			The top left corner of the array:
     1,     2,     6,     24,     120,      720,      5040,      40320
     3,     8,    30,    144,     840,     5760,     45360,     403200
    11,    38,   174,    984,    6600,    51120,    448560,    4394880
    49,   212,  1158,   7584,   57720,   499680,   4843440,   51932160
   261,  1370,  8742,  65304,  557400,  5343120,  56775600,  661933440
  1631, 10112, 74046, 622704, 5900520, 62118720, 718709040, 9059339520

Transpose: A276589.
Topmost row (row 0): A000142, Row 1: A001048 (without its initial 2), Row 2: A001344 (from a(1) = 11 onward), Row 3: A001345 (from a(1) = 49 onward), Row 4: A001346 (from a(1) = 261 onward), Row 5: A001347 (from a(1) = 1631 onward).
Leftmost column (column 0): A001339, Column 1: A001340, Columns 2-3: A001341 & A001342 (apparently).
Cf. A276075.
Cf. also arrays A066117, A276586, A099884, A255483.

T[r_, c_]:=Sum[Binomial[r, k](1 + c + k)!, {k, 0, r}]; Table[T[c, r - c], {r, 0, 10}, {c, 0, r}] // Flatten (* Indranil Ghosh, Apr 11 2017 *)

T(r, c) = sum(k=0, r, binomial(r, k)*(1 + c + k)!);
for(r=0, 10, for(c=0, r, print1(T(c, r - c),", ");); print();) \\ Indranil Ghosh, Apr 11 2017

from sympy import binomial, factorial
def T(r, c): return sum([binomial(r, k) * factorial(1 + c + k) for k in range(r + 1)])
for r in range(11): print([T(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Apr 11 2017

(define (A276588 n) (A276588bi (A002262 n) (A025581 n)))
(define (A276588bi row col) (A276075 (A066117bi (+ 1 row) (+ 1 col)))) ;; Code for A066117bi given in A066117, and for A276075 under the respective entry.

A(row,col) = Sum_{k=0..row} binomial(row,k)*A000142(1+col+k).

A(row,col) = A276075(A066117(row+1,col+1)).

A130478 Triangle T(n,k) = n! / A130477(n,k).

Original entry on oeis.org

1, 2, 2, 6, 3, 2, 24, 8, 3, 2, 120, 30, 8, 3, 2, 720, 144, 30, 8, 3, 2, 5040, 840, 144, 30, 8, 3, 2, 40320, 5760, 840, 144, 30, 8, 3, 2, 362880, 45360, 5760, 840, 144, 30, 8, 3, 2, 3628800, 403200, 45360, 5760, 840, 144, 30, 8, 3, 2

Offset: 1

Gary W. Adamson, May 31 2007

Sums of reciprocals of rows is 1. - Henry Bottomley, Nov 05 2009

			First few rows of the triangle:
     1;
     2,   2;
     6,   3,   2;
    24,   8,   3,  2;
   120,  30,   8,  3, 2;
   720, 144,  30,  8, 3, 2;
  5040, 840, 144, 30, 8, 3, 2;
  ...
Row 4 = (24, 8, 3, 2), terms such that (24, 8, 3, 2) dot (1, 3, 8, 12) = (24, 24, 24, 24), where (1, 3, 8, 12) = row 4 of A130477 and (24, 24, 24, 24) = row 4 of A130493.
Row 5 = (120, 30, 8, 3, 2) = 5! + (4!+3!) + (3!+2!) + (2!+1!) + (1!+1).
Row 5 = 120 followed by the first reversed 4 terms of A001048; i.e., 120 followed by 30, 8, 3, 2.

Cf. A130493 (row sums), A001048, A130493, A130477.

T(n,k) = n! / A130477(n,k); such that by rows as vector terms, (n-th row of A130477) dot (n-th row of A130478) = n-th row of A130493 = n! repeated n times.
Triangle by rows = n! followed by the first (n-1) reversed terms of A001048: (2, 3, 8, 30, 144, 840, ...).
Left border = (1, 2, 6, 24, 120, ...); while all other columns = A001048: (2, 3, 8, 30, ...).
n-th row of the triangle = n terms of: (n!; (n-1)!+(n-2)!; (n-2)!+(n-3)!; ...; 1! + 0!).

Corrected and extended by Henry Bottomley, Nov 05 2009

A052649 Expansion of e.g.f. (2+x-x^2)/(1-x)^2.

Original entry on oeis.org

2, 5, 14, 54, 264, 1560, 10800, 85680, 766080, 7620480, 83462400, 997920000, 12933043200, 180583603200, 2702527027200, 43153254144000, 732297646080000, 13160434839552000, 249692574523392000, 4987449116762112000, 104614786351595520000, 2299092397726924800000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(1) is 5 and gives the row number in the table of 0-origin permutations of order 3 in which the first 3 items are reversed. Row 5 of this table is 2 1 0. a(2) is 14 and gives the row number in the table of 0-origin permutations of order 4 in which the first three items are reversed. Row 14 of this table is 2 1 0 3.... a(6) is 10800 and gives the row number in the table of 0-origin permutations of order 8 in which the first 3 items are reversed. Row 10800 of this table is 2 1 0 3 4 5 6 7. Et cetera. - Eugene McDonnell (eemcd(AT)mac.com), Dec 03 2004

In factorial base representation (A007623) the terms of this sequence are written as: 10, 21, 210, 2100, 21000, 210000, ... From a(1) = 5 = "21" onward each term begins always with "21", which is then followed by n-1 zeros. - Antti Karttunen, Sep 24 2016

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 596.
Index entries for sequences related to factorial base representation.

Cf. A000142, A001048, A007623, A129326.

Row 4 of A276955 (from a(1)=5 onward).

spec := [S,{S=Prod(Sequence(Z),Union(Z,Sequence(Z),Sequence(Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

f[n_] := (3 + 2 n) n!; f[0] = 2; Array[f, 19, 0]
a[n_] := a[n] = a[n - 1]*n (2 n + 3)/(2 n + 1); a[0] = 2; a[1] = 5; Array[ a, 19, 0] (* Robert G. Wilson v *)
With[{nn=20},CoefficientList[Series[(2+x-x^2)/(1-x)^2,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Nov 09 2017 *)

a(n)=if(n<=1,[2,5][n+1], a(n-1)*(n*(2*n+3))/(2*n+1) );
for(n=0,11,print1(a(n),", "))

(define (A052649 n) (if (zero? n) 2 (+ (A000142 n) (* 2 (A000142 (+ 1 n)))))) ;; Antti Karttunen, Sep 24 2016

a(n) = (3+2*n)*n!.
E.g.f.: -(-x+x^2-2)/(-1+x)^2.
Recurrence: a(0)=2, a(1)=5, (-7*n-5-2*n^2)*a(n)+(3+2*n)*a(n+1)=0 for n>=1.
a(n) = A129326(n), n>1. - R. J. Mathar, Jun 14 2008
a(n) = (n+1)*a(n-1) - 2*A001048(n-1). - Gary Detlefs, Dec 16 2009
a(0) = 2; for n >= 1, a(n) = 2*(n+1)! + n! - Antti Karttunen, Sep 24 2016
From Amiram Eldar, Feb 17 2024: (Start)
Sum_{n>=0} 1/a(n) = 1/6 + e/2 - erfi(1)*sqrt(Pi)/4, where erfi is the imaginary error function.
Sum_{n>=0} (-1)^n/a(n) = 1/6 - 1/(2*e) + erf(1)*sqrt(Pi)/4, where erf is the error function. (End)

A052881 Expansion of e.g.f. log(1/(1-x))*x/(1-x).

Original entry on oeis.org

0, 0, 2, 9, 44, 250, 1644, 12348, 104544, 986256, 10265760, 116915040, 1446526080, 19323757440, 277238626560, 4251984710400, 69426608025600, 1202482800691200, 22021300630425600, 425162773111910400

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Matt Davis, Quadrant Marked Mesh Patterns and the r-Stirling Numbers, arXiv preprint arXiv:1412.0345 [math.CO], 2014 and J. Int. Seq. 18 (2015) 15.10.1.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 853

spec := [S,{B=Sequence(Z,1 <= card),C=Cycle(Z),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a:=n->abs(Stirling1(n,2))*n: seq(a(n), n=0..19); # Zerinvary Lajos, Oct 05 2007
A052881 := n -> `if`(n<2,0,2*(n-1)*(n-1)!*hypergeom([1,1,2-n],[2,n+1],-1)):
seq(simplify(A052881(n)),n=0..19); # Peter Luschny, Jun 11 2016

Table[n!*SeriesCoefficient[-Log[-1/(-1+x)]*x/(-1+x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 09 2012 *)
With[{nn=20},CoefficientList[Series[Log[1/(1-x)] x/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 19 2022 *)

x='x+O('x^66); concat([0,0],Vec(serlaplace(-log(-1/(-1+x))*x/(-1+x)))) \\ Joerg Arndt, May 06 2013

[stirling_number1(i,2)*i for i in range(0,32)] # Zerinvary Lajos, Jun 27 2008

E.g.f.: -log(-1/(-1+x))*x/(-1+x).
Recurrence: a(1)=0, a(2)=2, (n^3+3*n^2+2*n)*a(n)+(-5*n-2*n^2-2)*a(n+1)+(n+1)*a(n+2) =0.
a(n) = n!*Sum 1/i, i = 1..(n-1) = s(n, 2)-(n-1)! = n*s(n-1, 2) = n*a(n-1) + (n-1)! + (n-2)! = A000142(n)*A001008(n-1)/A002805(n-1) = A000254(n)-A000142(n-1) = A000027(n)*A000254(n-1) = a(n-1)*A000027(n) + A001048(n-1). - Henry Bottomley, May 05 2001
a(n) ~ n!*log(n)*(1+gamma/log(n)), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 09 2012
a(n) = 2*(n-1)*(n-1)!*hypergeom([1,1,2-n], [2,n+1], -1) for n>=2. - Peter Luschny, Jun 11 2016

New name using e.g.f., Vaclav Kotesovec, Feb 25 2014

A054115 Triangular array generated by its row sums: T(n,0)=1 for n >= 1, T(n,1)=r(n-1), T(n,k)=T(n,k-1)+r(n-k) for k=2,3,...,n, n >= 2, r(h)=sum of the numbers in row h of T.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 6, 8, 9, 1, 24, 30, 32, 33, 1, 120, 144, 150, 152, 153, 1, 720, 840, 864, 870, 872, 873, 1, 5040, 5760, 5880, 5904, 5910, 5912, 5913, 1, 40320, 45360, 46080, 46200, 46224, 46230, 46232, 46233, 1, 362880, 403200

Offset: 0

Clark Kimberling

			Rows:
1;
1,1;
1,2,3;
1,6,8,9;
1,24,30,32,33;
1,120,144,150,152,153;

n-th row sum is A000142(n+1) = (n+1)!.

T(n, n)=A007489(n) for n >= 1, T(n+1, 2)=A001048(n)

A076571 Binomial triangle based on factorials.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 6, 8, 11, 16, 24, 30, 38, 49, 65, 120, 144, 174, 212, 261, 326, 720, 840, 984, 1158, 1370, 1631, 1957, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 13700, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 109601

Offset: 0

Henry Bottomley, Oct 19 2002

			Rows start:
    1;
    1,   2;
    2,   3,   5;
    6,   8,  11,  16;
   24,  30,  38,  49,  65;
  120, 144, 174, 212, 261, 326;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35. See Table I.
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Columns include A000142, A001048, A001344, A001345, A001346, A001347.
Right hand columns include A000522, A001339, A001340, A001341, A001342.
Cf. A002627 (row sums), A099022.

A076571:= func< n,k| (&+[Binomial(k,j)*Factorial(n-j): j in [0..k]]) >;
[A076571(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

A076571[n_, k_]:= n!*Hypergeometric1F1[-k,-n,1];
Table[A076571[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 05 2023 *)

def A076571(n,k): return sum(binomial(k,j)*factorial(n-j) for j in range(k+1))
flatten([[A076571(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

T(n, k) = Sum_{j=0..k} binomial(k, j)*(n-j)!.
T(n, k) = T(n, k-1) + T(n-1, k-1) with T(n, 0) = n!.
T(n, n) = A000522(n).
Sum_{k=0..n} T(n, k) = A002627(n+1).
From G. C. Greubel, Oct 05 2023: (Start)
T(n, k) = n! * Hypergeometric1F1([-k], [-n], 1).
T(2*n, n) = A099022(n). (End)

A130744 a(n) = n(n+2)n!.

Original entry on oeis.org

0, 3, 16, 90, 576, 4200, 34560, 317520, 3225600, 35925120, 435456000, 5708102400, 80472268800, 1214269056000, 19527937228800, 333456963840000, 6025763487744000, 114887039275008000, 2304854534062080000

Offset: 0

Paul Curtz, Jul 12 2007

For n >= 1, a(n) = number whose factorial base representation (A007623) begins with a double digit {n}{n}, which is followed by n-1 zeros. Viewed in that base, this sequence looks like this: 0, 11, 220, 3300, 44000, 550000, 6600000, 77000000, 880000000, 9900000000, AA000000000, BB0000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - Antti Karttunen, May 07 2015

			G.f. = 3*x + 16*x^2 + 90*x^3 + 576*x^4 + 4200*x^5 + 34560*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for sequences related to factorial base representation.

Cf. A000142, A001048, A005563, A007623.
Column 3 of A257503 (apart from initial zero. Equally, row 3 of A257505).
Subsequence of both A227130 and A227148.
Cf. A001113, A001620, A091725, A099285.

[n*(n+2)*Factorial(n): n in [0..25]]; // Vincenzo Librandi, Aug 11 2011

a[n_]:=(n+2)!-(n+1)!-n!; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)

a(n)=n!*(n*(n+2)) \\ Charles R Greathouse IV, Aug 11 2011

(define (A130744 n) (* n (+ 2 n) (A000142 n))) ;; Antti Karttunen, May 07 2015

0 = +a(n) * (+a(n+1) + 2*a(n+2) - 6*a(n+3) + a(n+4)) + a(n+1) * (+5*a(n+2) - 6*a(n+3) + a(n+4)) + a(n+2) * (+3*a(n+2) - a(n+4)) + a(n+3) * (+a(n+3)) if n>=0. - Michael Somos, Mar 26 2014
From Antti Karttunen, May 07 2015: (Start)
a(n) = n * (n! + (n+1)!) = n * A001048(n+1).
a(n) = A005563(n) * A000142(n).
a(n) = (n+2)! - (n+1)! - n! [from Orlovsky's Mathematica-code].
(End)
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=1} 1/a(n) = (Ei(1) - gamma)/2 - 1/4, where Ei(1) = A091725 and gamma = A001620.
Sum_{n>=1} (-1)^(n+1)/a(n) = (gamma - Ei(-1))/2 - 1/e + 1/4, where Ei(-1) = -A099285 and e = A001113. (End)

More terms from Vladimir Joseph Stephan Orlovsky, Dec 05 2008

A054119 a(n) = n! + (n-1)! + (n-2)!.

Original entry on oeis.org

1, 2, 4, 9, 32, 150, 864, 5880, 46080, 408240, 4032000, 43908480, 522547200, 6745939200, 93884313600, 1401079680000, 22317642547200, 377917892352000, 6778983923712000, 128403161542656000, 2560949482291200000, 53645489280294912000, 1177524571957493760000, 27027108408834293760000

Offset: 0

Clark Kimberling

nonn

In factorial base representation (A007623) the terms are written as: 1, 10, 20, 111, 1110, 11100, 111000, ... From a(3) = 9 = "111" onward each term begins always with three consecutive 1's, followed by n-3 zeros. - Antti Karttunen, Sep 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Michele Battagliola, Nadir Murru, and Giordano Santilli, Combinatorial properties of multidimensional continued fractions, arXiv:2209.08875 [math.NT], 2022. See Proposition 3.3 pp. 6-7.
Index entries for sequences related to factorial base representation

Cf. A001048, A030495, A108217, A276940, A306770.
Equals T(n, 3), array T as in A054115.
Row 6 of A276955 (from a(3)=9 onward).

[1,2] cat [Factorial(n)+Factorial(n-1)+Factorial(n-2): n in [2..20]]; // Vincenzo Librandi, Oct 05 2011

f:= n-> `if`(n<0, 0, n!):
seq(f(n)+f(n-1)+f(n-2), n=0..23);

Join[{1,2},Table[n!+(n+1)!+(n+2)!,{n,0,30}]] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)
Join[{1,2,4},Plus@@@Partition[Range[30]!,3,1]] (* Harvey P. Dale, Aug 29 2024 *)

f(n) = if (n<0, 0, n!);
a(n) = f(n) + f(n-1) + f(n-2); \\ Michel Marcus, Sep 20 2022

(define (A054119 n) (if (<= n 1) (+ 1 n) (+ (A000142 n) (A000142 (- n 1)) (A000142 (- n 2))))) ;; Antti Karttunen, Sep 24 2016

For n>2, a(n) = (n-2)! * n^2. [Gary Detlefs, Aug 01 2009]
a(n) = (n+1)!*(H(n-1)+H(n+1)-H(n-2)-H(n))/2, n>1, where H(n) is the n-th harmonic number. [Gary Detlefs, Oct 04 2011]
E.g.f.: x + 1/(1-x) - x*log(1-x) = x^2/G(0)/2 where G(k) = 1 + (k+2)/(x - x*(k+1)/(x + k + 1 - x^4/(x^3 +(k+2)*(k+3)/G(k+1)))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Jul 06 2012
G.f.: G(0) where G(k) = 1 - x/(1 + x/(1 - x - (k+1)/( k+1 - x/Q))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Jul 28 2012
For n >= 1, a(n) = A276940(n)/n. - Antti Karttunen, Sep 24 2016
Sum_{n>=2} 1/a(n) = A306770. - Amiram Eldar, Nov 19 2020

Simpler definition from Miklos Kristof, Jun 16 2005

More terms from Antti Karttunen, Sep 24 2016

A166356 Expansion of e.g.f. 1 + x*arctanh(x), even powers only.

Original entry on oeis.org

1, 2, 8, 144, 5760, 403200, 43545600, 6706022400, 1394852659200, 376610217984000, 128047474114560000, 53523844179886080000, 26976017466662584320000, 16131658445064225423360000, 11292160911544957796352000000, 9146650338351415815045120000000

Offset: 0

Paul Barry, Oct 12 2009

For n>0, (4*n-1)*a(n) corresponds to the number of random walk labelings of the friendship graph F_n (i.e., the one-point union of n triangles). - Sela Fried, May 20 2023

Sela Fried and Toufik Mansour, Further results on random walk labelings, arXiv:2305.09971 [math.CO], 2023.

Cf. A053556, A001048, A068996, A143623.

a[0] = 1; a[n_] := (2*n - 1)! + (2*n - 2)!; Array[a, 14, 0] (* Amiram Eldar, Jan 02 2022 *)
With[{nn=40},Take[CoefficientList[Series[1+x ArcTanh[x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Apr 15 2024 *)

E.g.f.: 1 + x*arctanh(x) has expansion 1, 0, 2, 0, 8, 0, 144, ...
a(n) = (2n-1)! + (2n-2)! for n > 0; a(0) = 1.
a(n) -2*n*(2*n-3)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012
G.f.: 1 + x*G(0), where G(k) = 1 + 1/(1 - (k+2)*x/( (k+2)*x + (k+1)/((2*k+1)*(2*k+2))/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 08 2013
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=0} 1/a(n) = 2 - 1/e = 1 + A068996.
Sum_{n>=0} (-1)^n/a(n) = 2 - cos(1) - sin(1) = 2 - A143623. (End)

A261766 a(n) is the number of partial derangements of an n-set with at least one orbit of size exactly n.

Original entry on oeis.org

1, 0, 3, 8, 30, 144, 840, 5760, 45360, 403200, 3991680, 43545600, 518918400, 6706022400, 93405312000, 1394852659200, 22230464256000, 376610217984000, 6758061133824000, 128047474114560000, 2554547108585472000, 53523844179886080000, 1175091669949317120000

Offset: 0

Samira Stitou, Sep 21 2015

nonn

			a(3) = 8 because there are 8 partial derangements on {1,2,3} with at least one orbit of size 3 namely: (1,2) --> (2,3), (1,2)  --> (3,1), (1,3)  --> (2,1), (1,3) --> (3,2), (2,3)  --> (3,1), (2,3)  --> (1,2), (1,2,3) --> (2,3,1), (1,2,3)  --> (3,1,2).

A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.

Cf. A001048, A059171, A157400, A261762, A261763, A261764, A261765, A261767.

a(n) = A261765(n,n) - A261765(n,n-1) for n>0, a(0)=1.

More terms from Alois P. Heinz, Nov 04 2015

cp's OEIS Frontend

A276588 Square array A(row,col) = Sum_{k=0..row} binomial(row,k)*(1+col+k)!, read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

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A130478 Triangle T(n,k) = n! / A130477(n,k).

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A052649 Expansion of e.g.f. (2+x-x^2)/(1-x)^2.

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Maple

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A052881 Expansion of e.g.f. log(1/(1-x))*x/(1-x).

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A054115 Triangular array generated by its row sums: T(n,0)=1 for n >= 1, T(n,1)=r(n-1), T(n,k)=T(n,k-1)+r(n-k) for k=2,3,...,n, n >= 2, r(h)=sum of the numbers in row h of T.

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A076571 Binomial triangle based on factorials.

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Magma

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A130744 a(n) = n*(n+2)*n!.

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Magma

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A130744 a(n) = n(n+2)n!.