A008546 -id:A008546 - cp's OEIS frontend

A049382 Expansion of (1-25*x)^(-4/5).

1, 20, 450, 10500, 249375, 5985000, 144637500, 3512625000, 85620234375, 2092939062500, 51277007031250, 1258617445312500, 30941012197265625, 761624915625000000, 18768613992187500000, 462959145140625000000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

			(1-x)^(-4/5) = 1 + 4/5*x + 18/25*x^2 + 84/125*x^3 + ...

Cf. A008546, A034688, A049393, A049397, A049381, A049382.

A049382 := n -> (-25)^n*binomial(-4/5, n):
seq(A049382(n), n=0..16); # Peter Luschny, Oct 23 2018

CoefficientList[Series[(1-25x)^(-4/5),{x,0,20}],x] (* Harvey P. Dale, Oct 24 2021 *)

a(n) = prod(k=1, n, 25 - 5/k); \\ Michel Marcus, Jun 13 2018

G.f.: (1-25*x)^(-4/5).
a(n) = (5^n/n!) * Product_{k=0..n-1} (5*k + 4).
a(n) ~ Gamma(4/5)^-1*n^(-1/5)*5^(2*n)*{1 - 2/25*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = Product_{k=1..n} (25 - 5/k). - Michel Lagneau, Sep 16 2012
a(n) = (-25)^n*binomial(-4/5, n). - Peter Luschny, Oct 23 2018
From Peter Bala, Sep 24 2023: (Start)
a(n) = 25^n * binomial(n - 1/5, n).
P-recursive: a(n) = 5*(5*n - 1)/n * a(n-1) with a(0) = 1. (End)

A144828 Partial products of successive terms of A017113; a(0)=1.

Original entry on oeis.org

1, 4, 48, 960, 26880, 967680, 42577920, 2214051840, 132843110400, 9033331507200, 686533194547200, 57668788341964800, 5305528527460761600, 530552852746076160000, 57299708096576225280000, 6646766139202842132480000, 824199001261152424427520000, 108794268166472120024432640000

Offset: 0

Philippe Deléham, Sep 21 2008

a(n) is the number of signed permutations of length 4n that are equal to their reverse-inverses. Note that the reverse-inverse of a permutation is equivalent to a 90-degree rotation of the permutation's diagram (see the Hardt and Troyka link). - Justin M. Troyka, Aug 11 2011

Define the bar operation as an operation on signed permutation that flips the sign of each entry. Then a(n) is the number of signed permutations of length 2n that are equal to the bar of their inverses and equal to their reverse-complements (see the Hardt and Troyka link). - Justin M. Troyka, Aug 11 2011

			a(0)=1, a(1)=4, a(2)=4*12=48, a(3)=4*12*20=960, a(4)=4*12*20*28=26880, ...
Since a(1) = 4, there are 4 signed permutations of 4 that are equal to their reverse-inverses.  These are: (+2,+4,+1,+3), (+3,+1,+4,+2), (-2,-4,-1,-3), (-3,-1,-4,-2). - _Justin M. Troyka_, Aug 11 2011
G.f. = 1 + 4*x + 48*x^2 + 960*x^3 + 26880*x^4 + 967680*x^5 + 42577920*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179-217.
A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).

Cf. A000108, A001715, A002866, A007559, A008546, A017113, A047053, A048994, A049308, A144827.
Essentially the same as A052714. - N. J. A. Sloane, Feb 03 2013
Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), A052714 (or this sequence) (m=2), A221954 (m=3), A052734 (m=4), A221953 (m=5), A221955 (m=6).

[2^k *Factorial(2*k) / Factorial(k): k in [0..20]]; // Vincenzo Librandi, Aug 11 2011

A144828:= n-> 2^n*n!*binomial(2*n,n); seq(A144828(n), n=0..30); # G. C. Greubel, Apr 02 2021

Table[4^n (2 n - 1)!!, {n, 0, 15}] (* Vincenzo Librandi, May 14 2015 *)
Join[{1},FoldList[Times,(8*Range[0,20]+4)]] (* Harvey P. Dale, Dec 01 2015 *)

a(n)=binomial(2*n,n)*n!<Charles R Greathouse IV, Jan 17 2012

{a(n) = if( n<0, (-1)^n / a(-n), 2^n *(2*n)! / n!)}; /* Michael Somos, Jan 06 2017 */

[2^n*factorial(n+1)*catalan_number(n) for n in (0..30)] # G. C. Greubel, Apr 02 2021

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*8^(n-k).
a(n) = A052714(n+1). - R. J. Mathar, Oct 01 2008
a(n) = 2^n *(2*n)! / n!. - Justin M. Troyka, Aug 11 2011
G.f.: 1/(1-4x/(1-8x/(1-12x/(1-16x/(1-20x/(1-24x/(1-28x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (-4)^n*Sum_{k=0..n} 2^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
E.g.f.: 1/sqrt(1-8*x). - Philippe Deléham, May 14 2015
a(n) = 4^n * A001147(n). - Philippe Deléham, May 14 2015
a(n) = 8^n * Gamma(n + 1/2) / sqrt(Pi). - Daniel Suteu, Jan 06 2017
0 = a(n)*(8*a(n+1) - a(n+2)) + a(n+1)*(+a(n+1)) and a(n) = (-1)^n / a(-n) for all n in Z. - Michael Somos, Jan 06 2017
a(n) = 2^n * (n+1)! * Catalan(n). - G. C. Greubel, Apr 02 2021
Sum_{n>=0} 1/a(n) = 1 + e^(1/8)*sqrt(Pi)*erf(1/(2*sqrt(2)))/(2*sqrt(2)), where erf is the error function. - Amiram Eldar, Dec 20 2022

A011801 Triangle read by rows, the inverse Bell transform of n!*binomial(4,n) (without column 0).

Original entry on oeis.org

1, 4, 1, 36, 12, 1, 504, 192, 24, 1, 9576, 3960, 600, 40, 1, 229824, 100656, 17160, 1440, 60, 1, 6664896, 3048192, 563976, 54600, 2940, 84, 1, 226606464, 107255232, 21095424, 2256576, 142800, 5376, 112, 1, 8837652096, 4302305280, 887785920, 102332160, 7254576, 325584, 9072, 144, 1

Offset: 1

Wolfdieter Lang

Previous name was: Triangle of numbers related to triangle A049223; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n, m) = S2p(-4; n,m), a member of a sequence of triangles including S2p(-1; n,m) = A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) = A008277(n,m) (Stirling 2nd kind). T(n, 1) = A008546(n-1).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			Triangle starts:
          1;
          4,         1;
         36,        12,        1;
        504,       192,       24,       1;
       9576,      3960,      600,      40,      1;
     229824,    100656,    17160,    1440,     60,     1;
    6664896,   3048192,   563976,   54600,   2940,    84,    1;
  226606464, 107255232, 21095424, 2256576, 142800,  5376,  112,   1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Peter Luschny, The Bell transform
Index entries for sequences related to Bessel functions or polynomials

Cf. A008277, A008546, A049223, A264428.
Cf. A028575 (row sums).
Triangles with the recurrence T(n,k) = (m*(n-1)-k)*T(n-1,k) + T(n-1,k-1): A010054 (m=1), A001497 (m=2), A004747 (m=3), A000369 (m=4), this sequence (m=5), A013988 (m=6).

function T(n,k) // T = A011801
  if k eq 0 then return 0;
  elif k eq n then return 1;
  else return (5*(n-1)-k)*T(n-1,k) + T(n-1,k-1);
  end if;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2023

(* First program *)
T[n_, m_] /; n>=m>=1:= T[n, m]= (5*(n-1)-m)*T[n-1, m] + T[n-1, m-1]; T[n_, m_] /; nJean-François Alcover, Jun 20 2018 *)
(* Second program *)
rows = 10;
b[n_, m_]:= BellY[n, m, Table[k! Binomial[4, k], {k, 0, rows}]];
T= Table[b[n, m], {n,rows}, {m,rows}]//Inverse//Abs;
A011801= Table[T[[n, m]], {n,rows}, {m,n}]//Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses[inverse_bell_matrix from A264428]
# Adds 1,0,0,0, ... as column 0 at the left side of the triangle.
inverse_bell_matrix(lambda n: factorial(n)*binomial(4, n), 8) # Peter Luschny, Jan 16 2016

T(n, m) = n!*A049223(n, m)/(m!*5^(n-m)).
T(n+1, m) = (5*n-m)*T(n, m) + T(n, m-1), for n >= m >= 1, with T(n, m) = 0, for n < m, and T(n, 0) = 0, T(1, 1) = 1.
E.g.f. of n-th column: (1/n!)*( 1 - (1-5*x)^(1/5) )^n.
Sum_{k=1..n} T(n, k) = A028575(n).

New name from Peter Luschny, Jan 16 2016

A049210 a(n) = -Product_{k=0..n} (8*k-1); octo-factorial numbers.

Original entry on oeis.org

1, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425, 33760034483175, 2667042724170825, 232032717002861775, 22043108115271868625, 2270440135873002468375, 252018855081903273989625, 29990243754746489604765375, 3808760956852804179805202625

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..333
Index entries for sequences related to factorial numbers.

Cf. A034975, A048994, A051187.

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), this sequence (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

m:=8; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..30]]; // G. C. Greubel, Feb 16 2022

FoldList[Times,1,8*Range[20]-1] (* Harvey P. Dale, Aug 03 2014 *)
CoefficientList[Series[(1-8*x)^(-7/8),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

a(n) = -prod(k=0, n, 8*k-1); \\ Michel Marcus, Jan 08 2015

m=8; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022

a(n) = 7*A034975(n) = (8*n-1)(!^8), n >= 1, a(0) = 1.
G.f.: 1/(1-7*x/(1-8*x/(1-15*x/(1-16*x/(1-23*x/(1-24*x/(1-31*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (-1)^n*Sum_{k=0..n} 8^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 - 1/Q(0) )/x where Q(k) = 1 - x*(8*k-1)/(1 - x*(8*k+8)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
a(n) = 8^n*Gamma(n+7/8)/Gamma(7/8). - R. J. Mathar, Mar 20 2013
E.g.f: (1-8*x)^(-7/8). - Vaclav Kotesovec, Jan 28 2015
G.f.: 1/(1-7*x-56*x^2/(1-23*x-240*x^2/(1-39*x-552*x^2/(1-55*x-992*x^2/(1-71*x-1560*x^2/(1-... )))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Dec 09 2020
G.f.: 1/G(0) where G(k) = 1 - (16*k+7)*x - 8*(k+1)*(8*k+7)*x^2/G(k+1); (continued fraction). - Nikolaos Pantelidis, Dec 19 2020
Sum_{n>=0} 1/a(n) = 1 + (e/8)^(1/8)*(Gamma(7/8) - Gamma(7/8, 1/8)). - Amiram Eldar, Dec 20 2022

A144827 Partial products of successive terms of A017029; a(0)=1.

Original entry on oeis.org

1, 4, 44, 792, 19800, 633600, 24710400, 1136678400, 60243955200, 3614637312000, 242180699904000, 17921371792896000, 1451631115224576000, 127743538139762688000, 12135636123277455360000, 1237834884574300446720000, 134924002418598748692480000, 15651184280557454848327680000

Offset: 0

Philippe Deléham, Sep 21 2008

nonn

			a(0)=1, a(1)=4, a(2)=4*11=44, a(3)=4*11*18=792, a(4)=4*11*18*25=19800, ...

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A001715, A002866, A007559, A008546, A045754, A047053, A132393.

Cf. A049209, A049308, A051188, A084947, A144739, A147585.

[ 1 ] cat [ &*[ (7*k+4): k in [0..n] ]: n in [0..14] ]; // Klaus Brockhaus, Nov 10 2008

FoldList[Times,1,Range[4,150,7]] (* Harvey P. Dale, Apr 25 2014 *)

[1]+[4*7^(n-1)*rising_factorial(11/7, n-1) for n in (1..30)] # G. C. Greubel, Feb 22 2022

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*7^(n-k).
G.f.: 1/(1-4*x/(1-7*x/(1-11*x/(1-14*x/(1-18*x/(1-21*x/(1-25*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-3)^n*Sum_{k=0..n} (7/3)^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
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 7*x)^(4/7).
a(n) ~ sqrt(2*Pi)*7^n*n^(n+1/14)/(exp(n)*Gamma(4/7)). (End)
a(n) = 4*7^(n-1)*Pochhammer(n-1, 11/7) with a(0) = 1. - G. C. Greubel, Feb 22 2022
Sum_{n>=0} 1/a(n) = 1 + (e/7^3)^(1/7)*(Gamma(4/7) - Gamma(4/7, 1/7)). - Amiram Eldar, Dec 19 2022

Corrected a(9) by Vincenzo Librandi, Jul 14 2011

A157404 A partition product of Stirling_2 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 36, 1, 72, 144, 504, 1, 280, 1800, 2520, 9576, 1, 1740, 22320, 37800, 57456, 229824, 1, 8484, 182700, 864360, 1005480, 1608768, 6664896, 1, 57232, 2380896, 16546320, 26276544, 32175360, 53319168, 226606464

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144267.
Same partition product with length statistic is A011801.
Diagonal a(A000217) = A008546.
Row sum is A028575.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157405

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(5*j - 1).

A257614 Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), and f(n) = 5*n + 2.

Original entry on oeis.org

1, 2, 2, 4, 28, 4, 8, 244, 244, 8, 16, 1844, 5856, 1844, 16, 32, 13260, 101620, 101620, 13260, 32, 64, 93684, 1511160, 3455080, 1511160, 93684, 64, 128, 657836, 20663388, 91981880, 91981880, 20663388, 657836, 128, 256, 4609588, 269011408, 2121603436, 4047202720, 2121603436, 269011408, 4609588, 256

Offset: 0

Dale Gerdemann, May 09 2015

			Array t(n,k) begins as:
   1,      2,         4,           8,            16, ... A000079;
   2,     28,       244,        1844,         13260, ...;
   4,    244,      5856,      101620,       1511160, ...;
   8,   1844,    101620,     3455080,      91981880, ...;
  16,  13260,   1511160,    91981880,    4047202720, ...;
  32,  93684,  20663388,  2121603436,  146321752612, ...;
  64, 657836, 269011408, 44675623468, 4648698508440, ...;
Triangle T(n,k) begins as:
    1;
    2,      2;
    4,     28,        4;
    8,    244,      244,        8;
   16,   1844,     5856,     1844,       16;
   32,  13260,   101620,   101620,    13260,       32;
   64,  93684,  1511160,  3455080,  1511160,    93684,     64;
  128, 657836, 20663388, 91981880, 91981880, 20663388, 657836, 128;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A008546 (row sums), A142460, A257623.
Cf. A038208, A256890, A257609, A257610, A257612, A257616, A257617, A257618, A257619.
Similar sequences listed in A256890.

t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
T[n_, k_, p_, q_]= t[n-k, k, p, q];
Table[T[n,k,5,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 01 2022 *)

@CachedFunction
def t(n,k,p,q):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
def A257614(n,k): return t(n-k,k,5,2)
flatten([[A257614(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2022

T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 2.
Sum_{k=0..n} T(n, k) = A008546(n).
From G. C. Greubel, Mar 01 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000079(n). (End)

A051150 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -5, 1, 50, -15, 1, -750, 275, -30, 1, 15000, -6250, 875, -50, 1, -375000, 171250, -28125, 2125, -75, 1, 11250000, -5512500, 1015000, -91875, 4375, -105, 1, -393750000, 204187500, -41037500, 4230625

Offset: 1

Wolfdieter Lang

a(n,m) = R_n^m(a=0, b=5) in the notation of the given 1961 and 1962 references.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 5*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 5^d.

			Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
        1;
       -5,      1;
       50,    -15,      1;
     -750,    275,    -30,   1;
    15000,  -6250,    875, -50,    1;
  -375000, 171250, -28125, 2125, -75, 1;
  ...
3rd row o.g.f.: E(3,x) = 50*x - 15*x^2 + x^3.

Wolfdieter Lang, First 10 rows.
D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356. [The numbers R_n^m(a,b) are first introduced.]
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77. [Special cases of the numbers R_n^m(a,b) are tabulated.]

First (m=1) column sequence: A052562(n-1).
Row sums (signed triangle): A008546(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A008548(n).
Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4).

a(n, m) = a(n-1, m-1) - 5*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) := 0 for n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 5*x)/5)^m/m!.
a(n, m) = S1(n, m)*5^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

A254322 Expansion of e.g.f.: (1-11*x)^(-10/11).

Original entry on oeis.org

1, 10, 210, 6720, 288960, 15603840, 1014249600, 77082969600, 6706218355200, 657209398809600, 71635824470246400, 8596298936429568000, 1126115160672273408000, 159908352815462823936000, 24465977980765812062208000, 4012420388845593178202112000

Offset: 0

Vaclav Kotesovec, Jan 28 2015

Generally, for k > 1, if e.g.f. = (1-k*x)^(-(k-1)/k) then a(n) ~ n! * k^n / (n^(1/k) * Gamma((k-1)/k)).

G. C. Greubel, Table of n, a(n) for n = 0..300

Sequences of the form k^n*Pochhammer((k-1)/k, n): A000007 (k=1), A001147 (k=2), A008544 (k=3), A008545 (k=4), A008546 (k=5), A008543 (k=6), A049209 (k=7), A049210 (k=8), A049211 (k=9), A049212 (k=10), this sequence (k=11), A346896 (k=12).

Cf. A000108, A254282, A254286, A254287.

m=11; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022

CoefficientList[Series[(1-11*x)^(-10/11), {x, 0, 20}], x] * Range[0, 20]!
FullSimplify[Table[11^n * Gamma[n+10/11] / Gamma[10/11], {n, 0, 18}]]

m=11; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

D-finite with recurrence: a(0) = 1; a(n) = (11*n-1) * a(n-1) for n > 0. [corrected by Georg Fischer, Dec 23 2019]
a(n) = 11^n * Gamma(n+10/11) / Gamma(10/11).
a(n) ~ n! * 11^n / (n^(1/11) * Gamma(10/11)).
From Nikolaos Pantelidis, Jan 17 2021: (Start)
G.f.: 1/G(0) where G(k) = 1 - (22*k+10)*x - 11*(k+1)*(11*k+10)*x^2/G(k+1) (continued fraction).
G.f.: 1/(1-10*x-110*x^2/(1-32*x-462*x^2/(1-54*x-1056*x^2/(1-76*x-1892*x^2/(1-98*x-2970*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(11*k+10)/(1 - x*(11*k+11)/Q(k+1)) (continued fraction).
G.f.: 1/(1-10*x/(1-11*x/(1-21*x/(1-22*x/(1-32*x/(1-33*x/(1-43*x/(1-44*x/(1-54*x/(1-55*x/(1-...))))))))))) (Stieltjes continued fraction).
(End)
G.f.: hypergeometric2F0([1, 10/11], [--], 11*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/11)^(1/11)*(Gamma(10/11) - Gamma(10/11, 1/11)). - Amiram Eldar, Dec 22 2022

A144285 Lower triangular array called S2hat(-4) related to partition number array A144284.

Original entry on oeis.org

1, 4, 1, 36, 4, 1, 504, 52, 4, 1, 9576, 648, 52, 4, 1, 229824, 12888, 712, 52, 4, 1, 6664896, 286272, 13464, 712, 52, 4, 1, 226606464, 8182944, 299520, 13720, 712, 52, 4, 1, 8837652096, 266366016, 8455392, 301824, 13720, 712, 52, 4, 1, 388856692224, 10191545280, 273091392

Offset: 1

Wolfdieter Lang Oct 09 2008

If in the partition array M32khat(-4)= A144284 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-4). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.

The first three columns are A008546, A144339, A144340.

			[1];[4,1];[36,4,1];[504,52,4,1];[9576,648,52,4,1];...

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Row sums A144286.

A144280 (S2hat(-3)), A144342 (S2hat(-5)).

a(n,m)=sum(product(|S2(-4;j,1)|^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. |S2(-4,n,1)|= A011801(n,1) = A008546(n-1) = (5*n-6)(!^5) (5-factorials) for n>=2 and 1 if n=1.

cp's OEIS Frontend

A049382 Expansion of (1-25*x)^(-4/5).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A144828 Partial products of successive terms of A017113; a(0)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A011801 Triangle read by rows, the inverse Bell transform of n!*binomial(4,n) (without column 0).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A049210 a(n) = -Product_{k=0..n} (8*k-1); octo-factorial numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A144827 Partial products of successive terms of A017029; a(0)=1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A157404 A partition product of Stirling_2 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A257614 Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 2.

Views

Author

Keywords

Examples

A257614 Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), and f(n) = 5*n + 2.