A008545 -id:A008545 - cp's OEIS frontend

A225471 Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

1, 3, 1, 21, 10, 1, 231, 131, 21, 1, 3465, 2196, 446, 36, 1, 65835, 45189, 10670, 1130, 55, 1, 1514205, 1105182, 290599, 36660, 2395, 78, 1, 40883535, 31354119, 8951355, 1280419, 101325, 4501, 105, 1, 1267389585, 1012861224, 308846124, 48644344, 4421494, 240856, 7756, 136, 1

Offset: 0

Peter Luschny, May 17 2013

The Stirling-Frobenius cycle numbers are defined in A225470.

Triangle T(n,k), read by rows, given by (3, 4, 7, 8, 11, 12, 15, 16, 19, 20, ... (A014601)) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 14 2015

			[n\k][    0,       1,      2,     3,    4,  5,  6 ]
[0]       1,
[1]       3,       1,
[2]      21,      10,      1,
[3]     231,     131,     21,     1,
[4]    3465,    2196,    446,    36,    1,
[5]   65835,   45189,  10670,  1130,   55,  1,
[6] 1514205, 1105182, 290599, 36660, 2395, 78,  1.
...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence: T(4, 2) = T(3, 1) + (4*4 - 1)*T(3, 2) = 131 +15*21 = 446.
Boas-Buck recurrence for column k=2 and n=4: T(4, 2) = (4!/2)*(4*(3+8*(5/12)) *T(2, 2)/2! + 1*(3 + 8*(1/2))*T(3,2)/3!) = (4!/2)*(4*(19/3)/2  + 7*21/3!) =  446.
(End)

P. Bala, A 3 parameter family of generalized Stirling numbers.
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Columns k=0..3 give A008545, A286723(n-1), A383702, A383703.

Cf. A132393 (m=1), A028338 (m=2), A225470 (m=3).

T[0, 0] = 1; T[n_, k_] := Sum[Binomial[n - j, k]*Abs[StirlingS1[n, n - j]]* 3^(n - k - j)*4^j, {j, 0, n - k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018, after Wolfdieter Lang *)

@CachedFunction
def SF_C(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return SF_C(n-1, k-1, m) + (m*n-1)*SF_C(n-1, k, m)
for n in (0..8): [SF_C(n, k, 4) for k in (0..n)]

For a recurrence see the Sage program.
T(n, 0) ~ A008545; T(n, n) ~ A000012; T(n, n-1) = A014105.
Row sums ~ A047053; alternating row sums ~ A001813.
From Wolfdieter Lang, May 29 2017: (Start)
This is the Sheffer triangle (1/(1 - 4*x)^{-3/4}, -(1/4)*log(1-4*x)). See the P. Bala link where this is called exponential Riordan array, and the signed version is denoted by s_{(4,0,3)}.
E.g.f. of row polynomials in the variable x (i.e., of the triangle): (1 - 4*z)^{-(3+x)/4}.
E.g.f. of column k: (1-4*x)^(-3/4)*(-(1/4)*log(1-4*x))^k/k!, k >= 0.
Recurrence for row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k: R(n, x) = (x+3)*R(n-1,x+4), with R(0, x) = 1.
R(n, x) = risefac(4,3;x,n) := Product_{j=0..(n-1)} (x + (3 + 4*j)). (See the P. Bala link, eq. (16) for the signed s_{4,0,3} row polynomials.)
T(n, k) = Sum_{j=0..(n-m)} binomial(n-j, k)* S1p(n, n-j)*3^(n-k-j)*4^j, with S1p(n, m) = A132393(n, m).
T(n, k) = sigma[4,3]^{(n)}_{n-k}, with the elementary symmetric functions sigma[4,3]^{(n)}_m of degree m in the n numbers 3, 7, 11, ..., 3+4*(n-1), with sigma[4,3]^{(n)}_0 := 1. (End)
Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 4^(n-1-p)*(3 + 8*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, and beta(k) = A002208(k+1)/A002209(k+1), beginning with {1/2, 5/12, 3/8, 251/720, ...}. See a comment and references in A286718. - Wolfdieter Lang, Aug 11 2017

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

A303487 a(n) = n! * [x^n] 1/(1 - 4*x)^(n/4).

Original entry on oeis.org

1, 1, 12, 231, 6144, 208845, 8648640, 422463195, 23781703680, 1515973484025, 107941254220800, 8491022274509775, 731304510986649600, 68444451854354701125, 6916953288171902976000, 750681472158682148959875, 87076954662428278259712000, 10751175443940144673035200625

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

nonn

			a(1) = 1;
a(2) = 2*6 = 12;
a(3) = 3*7*11 = 231;
a(4) = 4*8*12*16 = 6144;
a(5) = 5*9*13*17*21 = 208845, etc.

Index entries for sequences related to factorial numbers

Column k=4 of A303489.

Cf. A000407, A001813, A007696, A008545, A034176, A034177, A047053, A051617, A051618, A051619, A051620, A051621, A051622, A113551, A303486, A303488.

Table[n! SeriesCoefficient[1/(1 - 4 x)^(n/4), {x, 0, n}], {n, 0, 17}]
Table[Product[4 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[4^n Pochhammer[n/4, n], {n, 0, 17}]

a(n) = Product_{k=0..n-1} (4*k + n).
a(n) = 4^n*Gamma(5*n/4)/Gamma(n/4).
a(n) ~ 5^(5*n/4-1/2)*n^n/exp(n).

A143173 Partition number array, called M32(-3), related to A000369(n,m) = |S2(-3;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 3, 1, 21, 9, 1, 231, 84, 27, 18, 1, 3465, 1155, 630, 210, 135, 30, 1, 65835, 20790, 10395, 4410, 3465, 3780, 405, 420, 405, 45, 1, 1514205, 460845, 218295, 169785, 72765, 72765, 30870, 19845, 8085, 13230, 2835, 735, 945, 63, 1, 40883535, 12113640, 5530140, 4074840

Offset: 1

Wolfdieter Lang, Oct 09 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k)=:M32(-3;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+2)-ary trees if the outdegree is r >= 0.
If M32(-3;n,k) is summed over those k with fixed number of parts m one obtains triangle A000369(n,m) = |S2(-3;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.

			a(4,3)=27. The relevant partition of 4 is (2^2). The 12 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are ternary because r=1 vertices are ternary (3-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4 versions due to the two ternary root vertices.

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.

Cf. A143172 (M32(-2) array), A144267 (M32(-4) array).

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-3,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-3,j,1)|^e(n,k,j),j=1..n), with |S2(-3,n,1)|= A008545(n-1) = (4*n-5)(!^4) (4-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).

A144280 Lower triangular array called S2hat(-3) related to partition number array A144279.

Original entry on oeis.org

1, 3, 1, 21, 3, 1, 231, 30, 3, 1, 3465, 294, 30, 3, 1, 65835, 4599, 321, 30, 3, 1, 1514205, 81081, 4788, 321, 30, 3, 1, 40883535, 1837836, 84483, 4869, 321, 30, 3, 1, 1267389585, 47609100, 1892835, 85050, 4869, 321, 30, 3, 1, 44358635475, 1449052605, 48681864

Offset: 1

Wolfdieter Lang, Oct 09 2008

If in the partition array M32khat(-3)= A144279 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-3). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.

The first three columns are A008545, A144282, A144283.

			Triangle begins:
  [1];
  [3,1];
  [21,3,1];
  [231,30,3,1];
  [3465,294,30,3,1];
  ...

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

Cf. A144279, A008277, A036040, A008284, A000369, A008545, A144282, A144283.
Row sums A144281.
Cf. A144275 (S2hat(-2)).

a(n,m) = Sum_{q=1..p(n,m)} Product_{j=1..n} |S2(-3;j,1)|^e(n,m,q,j) 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(-3,n,1)|= A000369(n,1) = A008545(n-1) = (4*n-5)(!^4) (4-factorials) for n>=2 and 1 if n=1.

A346896 Expansion of e.g.f.: (1-12*x)^(-11/12).

Original entry on oeis.org

1, 11, 253, 8855, 416185, 24554915, 1743398965, 144702114095, 13746700839025, 1470896989775675, 175036741783305325, 22929813173612997575, 3278963283826658653225, 508239308993132091249875, 84875964601853059238729125, 15192797663731697603732513375

Offset: 0

Nikolaos Pantelidis, Aug 06 2021

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

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), A049210 (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), this sequence (m=12).

m:=12; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 16 2022

CoefficientList[Series[(1-12*x)^(-11/12),{x,0,20}], x] * Range[0, 20]!
FullSimplify[Table[12^n Gamma[n+11/12]/Gamma[11/12],{n,0,15}]] (* Stefano Spezia, Aug 07 2021 *)

m=12; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 16 2022

G.f.: 1/(1-11*x/(1-12*x/(1-23*x/(1-24*x/(1-35*x/(1-36*x/(1-47*x/(1-48*x/(1-59*x/(1-60*x/(1-...))))))))))) (Stieltjes continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(12*k+11)/(1 - x*(12*k+12)/Q(k+1) ) (continued fraction).
G.f.: 1/(1-11*x-132*x^2/(1-35*x-552*x^2/(1-59*x-1260*x^2/(1-83*x-2256*x^2/(1-107*x-3540*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/G(0) where G(k) = 1 - x*(24*k+11) - 12*(k+1)*(12*k+11)*x^2/G(k+1) (continued fraction).
a(n) = 12^n*Gamma(n+11/12)/Gamma(11/12). - Stefano Spezia, Aug 07 2021
Sum_{n>=0} 1/a(n) = 1 + (e/12)^(1/12)*(Gamma(11/12) - Gamma(11/12, 1/12)). - Amiram Eldar, Dec 22 2022

A352073 Expansion of e.g.f. 1/(1 - log(1 + 4*x))^(1/4).

Original entry on oeis.org

1, 1, 1, 17, 1, 1889, -12415, 631665, -11224575, 461864385, -13754112255, 596055636945, -24148300842495, 1181210529292065, -59009709972278655, 3297137505670374705, -193318225258785780735, 12263541239089421903745, -820804950905249837195775

Offset: 0

Seiichi Manyama, Mar 05 2022

sign

Cf. A006252, A097397, A352070.

Cf. A007696, A008545, A352119.

m = 18; Range[0, m]! * CoefficientList[Series[(1 - Log[1 + 4*x])^(-1/4), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+4*x))^(1/4)))

a(n) = sum(k=0, n, 4^(n-k)*prod(j=0, k-1, 4*j+1)*stirling(n, k, 1));

a(n) = Sum_{k=0..n} 4^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-4)^k * (3/4 * k/n - 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 18 2023

A049410 A triangle of numbers related to triangle A049325.

Original entry on oeis.org

1, 3, 1, 6, 9, 1, 6, 51, 18, 1, 0, 210, 195, 30, 1, 0, 630, 1575, 525, 45, 1, 0, 1260, 10080, 6825, 1155, 63, 1, 0, 1260, 51660, 71505, 21840, 2226, 84, 1, 0, 0, 207900, 623700, 333585, 57456, 3906, 108, 1, 0, 0, 623700, 4573800, 4293135, 1195425, 131670

Offset: 1

Wolfdieter Lang

a(n,1)= A008279(3,n-1). a(n,m)=: S1(-3; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m)= A008275 (signed Stirling first kind), S1(2; n,m)= A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A000369(n,m).
The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
Also the inverse Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+3) (A008545) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

			Triangle begins:
  {1};
  {3,1};
  {6,9,1};
  {6,51,18,1};
  ...
E.g. row polynomial E(3,x)= 6*x+9*x^2+x^3.

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Row sums give A049426.

rows = 10;
t = Table[Product[4k+3, {k, 0, n-1}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
M = Inverse[Array[T, {rows, rows}]] // Abs;
A049325 = Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

# uses[inverse_bell_transform from A265605]
# Adds a column 1,0,0,0,... at the left side of the triangle.
multifact_4_3 = lambda n: prod(4*k + 3 for k in (0..n-1))
inverse_bell_matrix(multifact_4_3, 9) # Peter Luschny, Dec 31 2015

a(n, m) = n!*A049325(n, m)/(m!*4^(n-m)); a(n, m) = (4*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

A144758 Partial products of successive terms of A017197.

Original entry on oeis.org

1, 3, 36, 756, 22680, 884520, 42456960, 2420046720, 159723083520, 11979231264000, 1006255426176000, 93581754634368000, 9545338972705536000, 1059532625970314496000, 127143915116437739520000, 16401565050020468398080000, 2263415976902824638935040000

Offset: 0

Philippe Deléham, Sep 20 2008

			a(0)=1, a(1)=3, a(2)=3*12=36, a(3)=3*12*21=756, a(4)=3*12*21*30=22680, ...

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A048994, A132393.

Cf. A001710, A001147, A008545, A011781, A032031, A047056, A144739, A144756.

[Round(9^n*Gamma(n+1/3)/Gamma(1/3)): n in [0..20]]; // G. C. Greubel, Dec 03 2019

seq(9^n*pochhammer(1/3, n), n = 0..20); # G. C. Greubel, Dec 03 2019

Table[9^n*Pochhammer[1/3, n], {n, 0, 20}] (* G. C. Greubel, Dec 03 2019 *)
Join[{1},FoldList[Times,NestList[#+9&,3,20]]] (* Harvey P. Dale, Mar 09 2025 *)

a(n)=3^n*prod(i=1,n,3*i-2) \\ Charles R Greathouse IV, Jan 17 2012

[9^n*rising_factorial(1/3, n) for n in (0..20)] # G. C. Greubel, Dec 03 2019

a(n) = Sum_{k=0..n} A132393(n,k)*3^k*9^(n-k).
a(n) = (-6)^n*Sum_{k=0..n} (3/2)^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
Sum_{n>=0} 1/a(n) = 1 + (e/9^6)^(1/9)*(Gamma(1/3) - Gamma(1/3, 1/9)). - Amiram Eldar, Dec 21 2022

A254286 Expansion of (1 - (1-256x)^(1/4)) / (64x).

Original entry on oeis.org

1, 96, 14336, 2523136, 484442112, 98180268032, 20645907791872, 4459516083044352, 983075545417777152, 220208922173582082048, 49967406340478261526528, 11459191854083014643417088, 2651480699775516003646046208, 618173786004806016850049630208

Offset: 0

Vaclav Kotesovec, Jan 27 2015

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000108, A008545, A025749, A048779, A254282, A254287.

[Round(2^(8*n)*Gamma(n+3/4)/(Gamma(3/4)*Gamma(n+2))): n in [0..30]]; // G. C. Greubel, Aug 10 2022

CoefficientList[Series[(1-(1-256*x)^(1/4)) / (64*x),{x,0,20}],x]
CoefficientList[Series[Hypergeometric1F1[3/4,2,256*x],{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

[2^(8*n)*rising_factorial(3/4,n)/factorial(n+1) for n in (0..30)] # G. C. Greubel, Aug 10 2022

G.f.: (1 - (1-256*x)^(1/4)) / (64*x).
a(n) ~ 256^n / (Gamma(3/4) * n^(5/4)).
Recurrence: (n+1)*a(n) = 64*(4*n-1)*a(n-1).
a(n) = 256^n * Gamma(n+3/4) / (Gamma(3/4) * Gamma(n+2)).
E.g.f.: hypergeom([3/4], [2], 256*x). - Vaclav Kotesovec, Jan 28 2015
From Peter Bala, Sep 01 2017: (Start)
a(n) = (-1)^n*binomial(1/4, n+1)*4^(4*n+1). Cf. A000108(n) = (-1)^n*binomial(1/2, n+1)*2^(2*n+1).
a(n) = 16^n*A025749(n+1); a(n) = 32^n*A048779(n+1).
(End)

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cp's OEIS Frontend

A225471 Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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

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Magma

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Sage

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A303487 a(n) = n! * [x^n] 1/(1 - 4*x)^(n/4).

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A143173 Partition number array, called M32(-3), related to A000369(n,m) = |S2(-3;n,m)| (generalized Stirling triangle).

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A144280 Lower triangular array called S2hat(-3) related to partition number array A144279.

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A346896 Expansion of e.g.f.: (1-12*x)^(-11/12).

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A352073 Expansion of e.g.f. 1/(1 - log(1 + 4*x))^(1/4).

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PARI

PARI

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A049410 A triangle of numbers related to triangle A049325.

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A254286 Expansion of (1 - (1-256x)^(1/4)) / (64x).