A045755 -id:A045755 - cp's OEIS frontend

A144773 10-fold factorials: Product_{k=0..n-1} (10*k+1).

1, 1, 11, 231, 7161, 293601, 14973651, 913392711, 64850882481, 5252921480961, 478015854767451, 48279601331512551, 5359035747797893161, 648443325483545072481, 84946075638344404495011, 11977396665006561033796551, 1808586896415990716103279201, 291182490322974505292627951361

Offset: 0

Philippe Deléham, Sep 21 2008

G. C. Greubel, Table of n, a(n) for n = 0..320

Essentially a duplicate of A045757.
Cf. k-fold factorials: A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A256268 (combined table).
Cf. A048994, A132393.

R:=PowerSeriesRing(Rationals(), 15); Coefficients(R!(Laplace( (1-10*x)^(-1/10) ))); // G. C. Greubel, Mar 03 2020

G(x):=(1-10*x)^(-1/10): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..14); # Zerinvary Lajos, Apr 03 2009

b = 10; Table[FullSimplify[b^n*Gamma[n + 1/b]/Gamma[1/b]], {n, 0, 14}] (* Michael De Vlieger, Sep 14 2016 *)
Join[{1},FoldList[Times,10 Range[0,15]+1]] (* Harvey P. Dale, Oct 24 2022 *)

Vec(serlaplace( (1-10*x)^(-1/10) +O('x^15) )) \\ G. C. Greubel, Mar 03 2020

[10^n*rising_factorial(1/10,n) for n in (0..15)] # G. C. Greubel, Mar 03 2020

a(n) = Sum_{k = 0..n} (-10)^(n - k) * A048994(n, k).
a(n) = Sum_{k = 0..n} 10^(n - k) * A132393(n, k).
E.g.f.: (1 - 10*x)^(-1/10).
a(n) = A045757(n), n>0.
a(n) = (-9)^n * Sum_{k = 0..n} (10/9)^k * s(n + 1,n + 1 - k), where s(n, k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0), where Q(k) = 1 - (10*k+1)*x/( 1 - 10*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014
a(n) = 10^n * Gamma(n + 1/10) / Gamma(1/10). - Artur Jasinski Aug 23 2016
a(n) ~ sqrt(2*Pi)*10^n*n^(n-2/5)/(Gamma(1/10)*exp(n)). - Ilya Gutkovskiy, Sep 11 2016
D-finite with recurrence: a(n) - (10*n-9)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020
Sum_{n>=0} 1/a(n) = 1 + (e/10^9)^(1/10)*(Gamma(1/10) - Gamma(1/10, 1/10)). - Amiram Eldar, Dec 22 2022

A053115 a(n) = ((8n+10)(!^8))/20, related to A034908 ((8n+2)(!^8) octo- or 8-factorials).

Original entry on oeis.org

1, 18, 468, 15912, 668304, 33415200, 1938081600, 127913385600, 9465590534400, 776178423820800, 69856058143872000, 6845893698099456000, 725664731998542336000, 82725779447833826304000

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(9; m,n) := ((8*n+m)(!^8))/m(!^8), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..332

Cf. A051189, A045755, A034908-12, A034975-6, A053114 (rows m=0..9).

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-8*x)^(9/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 26 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 17, 5!, 8}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nmax = 50}, CoefficientList[Series[1/(1 - 8*x)^(9/4), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Aug 26 2018 *)

x='x+O('x^25); Vec(serlaplace(1/(1-8*x)^(9/4))) \\ G. C. Greubel, Aug 26 2018

a(n) = ((8*n+10)(!^8))/10(!^8) = A034908(n+2)/10.
E.g.f.: 1/(1-8*x)^(9/4).
G.f.: 1/(1-18x/(1-8x/(1-26x/(1-16x/(1-34x/(1-24x/(1-42x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012

A088996 Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 7, 6, 0, 6, 29, 46, 24, 0, 24, 146, 329, 326, 120, 0, 120, 874, 2521, 3604, 2556, 720, 0, 720, 6084, 21244, 39271, 40564, 22212, 5040, 0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320

Offset: 0

Philippe Deléham, Dec 01 2003, Aug 17 2007

			Triangle begins:
  1;
  0,    1;
  0,    1,     2;
  0,    2,     7,      6;
  0,    6,    29,     46,     24;
  0,   24,   146,    329,    326,    120;
  0,  120,   874,   2521,   3604,   2556,    720;
  0,  720,  6084,  21244,  39271,  40564,  22212,   5040;
  0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Trevor Hyde, Liminal reciprocity and factorization statistics, arXiv:1803.08438 [math.NT], 2018.

Variant: A059364, diagonals give A000007, A000142, A067318.

Cf. A001147 (row sums), A048994, A084938.

A088996:= func< n,k | (&+[(-1)^j*Binomial(j,n-k)*StirlingFirst(n,n-j): j in [0..n]]) >;
[A088996(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 23 2022

A059364 := (n, k) -> add(abs(Stirling1(n, n - j))*binomial(j, n - k), j = 0..n);
seq(seq(A059364(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Aug 27 2025

T[n_, k_]:= T[n, k]= Sum[(-1)^(n-i)*Binomial[i, k] StirlingS1[n+1, n+1-i], {i, 0, n}]; {{1}}~Join~Table[Abs@ T[n, k], {n,0,10}, {k,n+1,0,-1}] (* Michael De Vlieger, Jun 19 2018 *)

def A088996(n,k): return add((-1)^(n-i)*binomial(i,k)*stirling_number1(n+1,n+1-i) for i in (0..n))
for n in (0..10): [A088996(n,k) for k in (0..n)]  # Peter Luschny, May 12 2013

T(n, k) given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938. [Original name.]
Sum_{k=0..n} (-1)^k*T(n,k) = (-1)^n.
From Vladeta Jovovic, Dec 15 2004: (Start)
E.g.f.: (1-y-y*x)^(-1/(1+x)).
Sum_{k=0..n} T(n, k)*x^k = Product_{k=1..n} (k*x+k-1). (End)
T(n, k) = n*T(n-1, k-1) + (n-1)*T(n-1, k); T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0. - Philippe Deléham, May 22 2005
Sum_{k=0..n} T(n,k)*x^(n-k) = A019590(n+1), A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, respectively. Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000007(n), A001147(n), A008544(n), A008545(n), A008546(n), A008543(n), A049209(n), A049210(n), A049211(n), A049212(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Aug 10 2007
T(n, k) = Sum_{j=0..n} (-1)^j*binomial(j, n-k)*StirlingS1(n, n-j). - G. C. Greubel, Feb 23 2022

New name using a formula of G. C. Greubel by Peter Luschny, Aug 27 2025

A147626 Octo-factorial numbers (5).

Original entry on oeis.org

1, 6, 84, 1848, 55440, 2106720, 96909120, 5233092480, 324451733760, 22711621363200, 1771506466329600, 152349556104345600, 14320858273808486400, 1460727543928465612800, 160680029832131217408000, 18960243520191483654144000, 2388990683544126940422144000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..325

Cf. A034908, A045755, A048994, A049210, A051189, A052714, A132393, A147625.

[n le 1 select 1 else (8*n-10)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022

s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,5,2*5!,8}];lst
Table[8^(n-1)*Pochhammer[3/4, n-1], {n,40}] (* G. C. Greubel, Oct 21 2022 *)

[8^(n-1)*rising_factorial(3/4, n-1) for n in range(1,40)] # G. C. Greubel, Oct 21 2022

a(n+1) = Sum_{k=0..n} A132393(n,k)*6^k*8^(n-k). - Philippe Deléham, Nov 09 2008
a(n) = (-2)^n*Sum_{k=0..n} 4^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 2*x/G(0), where G(k) = 1 + 1/(1 - 2*x*(8*k+6)/(2*x*(8*k+6) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
From G. C. Greubel, Oct 21 2022: (Start)
a(n) = 8^n * Pochhammer(n, 3/4) = -2^(3*n-1) * Pochhammer(n, -1/4).
a(n) = (8*n - 10)*a(n-1). (End)
Sum_{n>=1} 1/a(n) = 1 + (e/8^2)^(1/8)*(Gamma(3/4) - Gamma(3/4, 1/8)). - Amiram Eldar, Dec 20 2022

A153274 Triangle, read by rows, T(n,k) = k^(n+1) * Pochhammer(1/k, n+1).

Original entry on oeis.org

2, 6, 15, 24, 105, 280, 120, 945, 3640, 9945, 720, 10395, 58240, 208845, 576576, 5040, 135135, 1106560, 5221125, 17873856, 49579075, 40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000, 362880, 34459425, 608608000, 4996616625, 26381811456, 104463111025, 337767408000, 939536222625

Offset: 1

Roger L. Bagula, Dec 22 2008

A Pochhammer function-based triangular sequence.

Row sums are: {2, 21, 409, 14650, 854776, 73920791, 8878927331, 1413788600036, 288152651134776, 73152069870215127, ...}.

			Triangle begins as:
      2;
      6,      15;
     24,     105,      280;
    120,     945,     3640,      9945;
    720,   10395,    58240,    208845,    576576;
   5040,  135135,  1106560,   5221125,  17873856,   49579075;
  40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000;

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755, A045756.

Cf. A142589, A256268.

Flat(List([1..12], n-> List([1..n], k-> Product([0..n], j-> j*k+1 )))); # G. C. Greubel, Mar 05 2020

[(&*[j*k+1: j in [0..n]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 05 2020

seq(seq( k^(n+1)*pochhammer(1/k, n+1), k=1..n), n=1..12); # G. C. Greubel, Mar 05 2020

Table[Apply[Plus, CoefficientList[j*k^n*Pochhammer[(j+k)/k, n], j]], {n, 12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 05 2020 *)
Table[k^(n+1)*Pochhammer[1/k, n+1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 05 2020 *)

T(n, k) = prod(j=0, n, j*k+1);
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Mar 05 2020

[[k^(n+1)*rising_factorial(1/k,n+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 05 2020

T(n, k) = k^(n+1) * Pochmammer(1/k, n+1).

T(n, k) = Product_{j=0..n} (j*k + 1). - G. C. Greubel, Mar 05 2020

Edited by G. C. Greubel, Mar 05 2020

A113149 Row 8 of table A113143; equal to INVERT of 8-fold factorials shifted one place right.

Original entry on oeis.org

1, 1, 2, 12, 176, 4184, 134824, 5451528, 264710536, 14992543432, 969925065992, 70547721068232, 5697913588192520, 505926926171909576, 48979597517592503560, 5134435963996172979912, 579379155027833982679816

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 28 2005

nonn

			A(x) = 1 + x + 2*x^2 + 12*x^3 + 176*x^4 + 4184*x^5 +...
= 1/(1 - x - x^2 - 9*x^3 - 153*x^4 -...- A045755(n)*x^(n+1)
-...).

Cf. A113143, A045755 (8-fold factorials).

{a(n)=local(x=X+X*O(X^n)); A=1/(1-x-x^2*sum(j=0,n,x^j*prod(i=0,j,8*i+1)));return(polcoeff(A,n,X))}

a(n) = Sum_{j=0..k} 8^(k-j)*A111146(k, j).

a(0) = 1; a(n+1) = Sum_{k=0..n} a(k)*A045755(n-k).

A153189 Triangle T(n,k) = Product_{j=0..k} n*j+1.

Original entry on oeis.org

1, 1, 2, 1, 3, 15, 1, 4, 28, 280, 1, 5, 45, 585, 9945, 1, 6, 66, 1056, 22176, 576576, 1, 7, 91, 1729, 43225, 1339975, 49579075, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625

Offset: 0

Roger L. Bagula, Dec 20 2008

Row sums are: {1, 3, 19, 313, 10581, 599881, 50964103, 6047094369, 954249517513, 193146844030201, 48762935887310811,...}. [Corrected by M. F. Hasler, Oct 28 2014]
This is the lower left triangle of the array A142589. - M. F. Hasler, Oct 28 2014
Row n is a subset of the n-fold factorial sequence for k=0..n. For example, T(8,0..8) is A045755(1..9).  These sequences are listed for n=0..10 in A256268. - Georg Fischer, Feb 15 2020

			Triangle begins as:
  1;
  1, 2;
  1, 3,  15;
  1, 4,  28,  280;
  1, 5,  45,  585,   9945;
  1, 6,  66, 1056,  22176,  576576;
  1, 7,  91, 1729,  43225, 1339975,  49579075;
  1, 8, 120, 2640,  76560, 2756160, 118514880,  5925744000;
  1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000165, A006882, A007661, A007662, A008544.

Cf. A000142 (row 2), A001147 (3), A007559 (4), A007696 (5), A008548 (6), A008542 (7), A045754 (8), A045755 (9), A045756 (10), A144773 (11), A256268 (combined table).

[(&*[n*j+1: j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 15 2020

seq(seq(mul(n*j+1, j=0..k), k=0..n), n=0..10); # G. C. Greubel, Feb 15 2020

T[n_, k_]= If[n==0 && k==0, 1, Product[n*j+1, {j,0,k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 15 2020 *)
T[n_, k_]:= T[n, k]= If[k<2, 1+k*n, ((1+n*k)*T[n, k-1] + (1+n*k)*(1+n*(k-1))* T[n, k-2])/2]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Georg Fischer, Feb 17 2020 *)

T(n,k)=prod(j=0,k,n*j+1) \\ M. F. Hasler, Oct 28 2014

[[ product(n*j+1 for j in (0..k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 15 2020

T(n, k) = n^(k+1)*Pochhammer(1/n, k+1).
From Vaclav Kotesovec, Oct 10 2016: (Start)
For fixed n > 0:
T(n, k) ~ sqrt(2*Pi) * n^k * k^(k + 1/2 + 1/n) / (Gamma(1 + 1/n) * exp(k)).
T(n, k) ~ k! * n^k * k^(1/n) / Gamma(1 + 1/n).
(End)
T(n, k) = Sum_{j=0..k+1} (-1)^(k-j+1)*Stirling1(k+1,j)*n^(k-j+1). - G. C. Greubel, Feb 17 2020
T(n, k) = ((1+n*k)*T(n, k-1) + (1+n*k)*(1+n*(k-1))*T(n, k-2))/2. - Georg Fischer, Feb 17 2020

Edited and row 0 added by M. F. Hasler, Oct 28 2014

A020028 Nearest integer to Gamma(n + 1/8)/Gamma(1/8).

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 20, 121, 862, 7000, 63876, 646743, 7195013, 87239530, 1145018832, 16173391002, 244622538903, 3944538439806, 67550220781672, 1224347751667801, 23415650750646696, 471239971356764754

Offset: 0

Simon Plouffe

nonn

Gamma(n + 1/8)/Gamma(1/8) = 1, 1/8, 9/64, 153/512, 3825/4096, 126225/32768, 5175225/262144, 253586025/2097152, ... - R. J. Mathar, Sep 04 2016

Cf. A045755, A020073, A020118.

Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;

Table[Round[Gamma[n+1/8]/Gamma[1/8]],{n,0,30}] (* Harvey P. Dale, Sep 05 2020 *)

A368119 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} (n*j + 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 15, 24, 1, 1, 1, 5, 28, 105, 120, 1, 1, 1, 6, 45, 280, 945, 720, 1, 1, 1, 7, 66, 585, 3640, 10395, 5040, 1, 1, 1, 8, 91, 1056, 9945, 58240, 135135, 40320, 1, 1, 1, 9, 120, 1729, 22176, 208845, 1106560, 2027025, 362880, 1

Offset: 0

Peter Luschny, Dec 18 2023

A(n, k) is the number of increasing (n + 1)-ary trees on k vertices. (Following a comment of David Callan in A007559.)

			Array A(n, k) starts:
  [0] 1, 1, 1,   1,    1,      1,       1,         1, ...  A000012
  [1] 1, 1, 2,   6,   24,    120,     720,      5040, ...  A000142
  [2] 1, 1, 3,  15,  105,    945,   10395,    135135, ...  A001147
  [3] 1, 1, 4,  28,  280,   3640,   58240,   1106560, ...  A007559
  [4] 1, 1, 5,  45,  585,   9945,  208845,   5221125, ...  A007696
  [5] 1, 1, 6,  66, 1056,  22176,  576576,  17873856, ...  A008548
  [6] 1, 1, 7,  91, 1729,  43225, 1339975,  49579075, ...  A008542
  [7] 1, 1, 8, 120, 2640,  76560, 2756160, 118514880, ...  A045754
  [8] 1, 1, 9, 153, 3825, 126225, 5175225, 253586025, ...  A045755

Index entries for sequences related to factorial numbers.

Rows: A000012, A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755.
Columns: A000012, A000027, A000384, A011199, A011245.
Variant: A256268. Main diagonal: A092985.
Cf. A124320, A008279, A326323.

def A(n, k): return n**k * rising_factorial(1/n, k) if n > 0 else 1
for n in range(9): print([A(n, k) for k in range(8)])

Let rf(n, k) denote the rising factorial and ff(n,k) the falling factorial.
A(n, k) = n^k * rf(1/n, k) if n > 0 else 1.
A(n, k) = (-n)^k * ff(-1/n, k) if n > 0 else 1.
A(n, k) = (n^k * Gamma(k + 1/n)) / Gamma(1/n) for n > 0.
A(n, k) = ((-n)^k * Gamma(1 - 1/n)) / Gamma(1 - 1/n - k) for n > 0.
A(n, k) = k! * [x^k](1 - n*x)^(-1/n).
A(n, k) = [x^k] hypergeom([1, 1/n], [], n*x).
Column n + 1 has a linear recurrence with constant coefficients and signature ((-1)^k*binomial(n+1, n-k) for k=0..n).

cp's OEIS Frontend

A144773 10-fold factorials: Product_{k=0..n-1} (10*k+1).

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A053115 a(n) = ((8*n+10)(!^8))/20, related to A034908 ((8*n+2)(!^8) octo- or 8-factorials).

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A088996 Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.

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A147626 Octo-factorial numbers (5).

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A153274 Triangle, read by rows, T(n,k) = k^(n+1) * Pochhammer(1/k, n+1).

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A113149 Row 8 of table A113143; equal to INVERT of 8-fold factorials shifted one place right.

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A153189 Triangle T(n,k) = Product_{j=0..k} n*j+1.

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A053115 a(n) = ((8n+10)(!^8))/20, related to A034908 ((8n+2)(!^8) octo- or 8-factorials).