A028575 -id:A028575 - cp's OEIS frontend

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

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

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).

A015735 Row sums of triangle A004747.

Original entry on oeis.org

1, 3, 17, 145, 1661, 23931, 415773, 8460257, 197360985, 5192853011, 152137882601, 4911873672113, 173268075672277, 6630323916472075, 273555262963272501, 12105084133976359361, 571897644855277242673, 28731255563712689630627, 1529450942687399074134465

Offset: 1

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A001515, A004747, A016036, A028575.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-3*x)^(1/3)) - 1 ))); // G. C. Greubel, Oct 02 2023

a[1]=1; a[n_]:= 1 +(n-1)!*Sum[Binomial[k, n-m-k]*Binomial[k+n-1,n-1]*(-1/3)^(n-m-k)/(m-1)!, {m,n}, {k,n-m}]; Table[a[n], {n,20}] (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)
Rest@With[{m=30}, CoefficientList[Series[Exp[1-Surd[1-3*x,3]] -1, {x, 0,m}], x]*Range[0,m]!] (* G. C. Greubel, Oct 02 2023 *)

a(n):=if n=1 then 1 else (n-1)!*sum(sum(binomial(k,n-m-k)* (-1/3)^(n-m-k)*binomial(k+n-1,n-1),k,1,n-m)/(m-1)!,m,1,n)+1; /* Vladimir Kruchinin, Aug 08 2010 */

def A015735_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-3*x)^(1/3)) -1 ).egf_to_ogf().list()
a=A015735_list(40); a[1:] # G. C. Greubel, Oct 02 2023

E.g.f.: exp(1-(1-3*x)^(1/3)) - 1, if one takes a(0)=0.
a(n) = 6*(n-2)*a(n-1) - (3*n-8)*(3*n-7)*a(n-2) + a(n-3), a(0)=1, a(1)=1, a(2)=3.
a(n) = 1 + (n-1)!*Sum_{m=1..n} ( Sum_{k=1..n-m} C(k, n-m-k)*C(k+n-1, n-1)*(-1/3)^(n-m-k) ) / (m-1)!, n > 1. - Vladimir Kruchinin, Aug 08 2010
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)^2*d/dx. Cf. A001515, A016036 and A028575. - Peter Bala, Nov 25 2011
E.g.f. with offset 0: exp(1-(1-3*x)^(1/3))/(1-3*x)^(2/3). - Sergei N. Gladkovskii, Jul 07 2012.
a(n) ~ sqrt(2*Pi)*3^(n-1)*exp(1-n)*n^(n-5/6)/Gamma(2/3) * (1-sqrt(3)*Gamma(2/3)^2/(2*Pi*n^(1/3))). - Vaclav Kotesovec, Aug 10 2013
From Seiichi Manyama, Jan 20 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 3^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-3)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/3,n)/k!. (End)

A016036 Row sums of triangle A000369.

Original entry on oeis.org

1, 4, 31, 361, 5626, 109951, 2585269, 71066626, 2236441141, 79289379361, 3127129674736, 135802922499949, 6439320471558781, 331026965612789356, 18338413238239145731, 1089132347371148170381, 69033182553940825258594, 4651256393180943757676371

Offset: 1

Wolfdieter Lang

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Sequences with e.g.f. exp(1-(1-m*x)^(1/m)) - 1: A000012 (m=1), A001515 (m=2), A015735 (m=3), this sequence (m=4), A028575 (m=5), A028844 (m=6).

Cf. A000369.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-4*x)^(1/4)) -1 ))); // G. C. Greubel, Oct 02 2023

a[n_, m_] /; (n>= m>= 1):= a[n, m]= (4*(n-1)-m)*a[n-1,m] + a[n-1,m-1]; a[n_, m_] /; n,0]= 0; a[1,1] = 1; a[n]:= Sum[a[n,m], {m, n}]; Table[a[n], {n,20}] (* Jean-François Alcover, Feb 28 2013 *)
With[{nn=20},CoefficientList[Series[Exp[1-Surd[1-4x,4]]-1,{x,0,nn}],x] Range[0,nn]!]//Rest (* Harvey P. Dale, Apr 20 2016 *)

a(n):=((n-1)!*sum((sum(binomial(n+k-1,n-1)*sum(binomial(j,n-m-3*k+2*j)*binomial(k,j)*3^(-n+m+3*k-j)*2^(n-m-5*k+3*j)*(-1)^(n-m-k),j,0,k),k,1,n-m))/(m-1)!,m,1,n-1))+1; /* Vladimir Kruchinin, Oct 18 2011 */

def A016036_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-4*x)^(1/4)) -1 ).egf_to_ogf().list()
a=A016036_list(40); a[1:] # G. C. Greubel, Oct 02 2023

E.g.f.: exp(1 - (1-4*x)^(1/4)) - 1.
a(n) = 6*(2*n-5)*a(n-1) - 3*(16*n^2-96*n+145)*a(n-2) + 2*(4*n-15)*(2*n-7)*(4*n-13)*a(n-3) + a(n-4), n >= 4; a(0) = 1, a(1) = 1, a(2) = 4, a(3) = 31.
a(n) = 1 + (n-1)!*Sum_{m=1..n-1} ( Sum_{k=1..n-m} binomial(n+k-1,n-1) * ( Sum_{j=0..k} binomial(j,n-m-3*k+2*j)*binomial(k,j)*3^(-n+m+3*k-j)*2^(n-m-5*k+3*j)*(-1)^(n-m-k) ) )/(m-1)!. - Vladimir Kruchinin, Oct 18 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)^3*d/dx. Cf. A001515, A015735 and A028575. - Peter Bala, Nov 25 2011
a(n) ~ 2^(2*n-3/2)*n^(n-3/4)*exp(1-n)*sqrt(Pi)/Gamma(3/4) * (1 - Gamma(3/4)/(n^(1/4)*sqrt(Pi)) + Gamma(3/4)^2/(4*sqrt(n/2)*Pi)). - Vaclav Kotesovec, Aug 10 2013
From Seiichi Manyama, Jan 20 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 4^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-4)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/4,n)/k!. (End)

A028844 Row sums of triangle A013988.

Original entry on oeis.org

1, 6, 71, 1261, 29906, 887751, 31657851, 1318279586, 62783681421, 3365947782611, 200610405843926, 13157941480889921, 941848076798467801, 73060842413607398806, 6105266987293752470991, 546770299628690541571901, 52244284936267317229542466, 5305131708827069245129523591

Offset: 1

Wolfdieter Lang

nonn

G. C. Greubel, Table of n, a(n) for n = 1..250
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Sequences with e.g.f. exp(1-(1-m*x)^(1/m)) - 1: A000012 (m=1), A001515 (m=2), A015735 (m=3), A016036 (m=4), A028575 (m=5), this sequence (m=6).

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-6*x)^(1/6)) -1 ))); // G. C. Greubel, Oct 03 2023

With[{nn=20},Rest[CoefficientList[Series[Exp[1-(1-6x)^(1/6)]-1,{x,0,nn}], x]Range[0,nn]!]] (* Harvey P. Dale, Feb 02 2012 *)

def A028844_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-6*x)^(1/6)) -1 ).egf_to_ogf().list()
a=A028844_list(40); a[1:] # G. C. Greubel, Oct 03 2023

E.g.f.: exp(1 - (1-6*x)^(1/6)) - 1.
D-finite with recurrence: a(n) = 15*(2*n-7)*a(n-1) +5*(72*n^2-576*n+1169)*a(n-2) +45*(2*n-9)*(24*n^2-216*n+497)*a(n-3) -20*(324*n^4-6480*n^3+48735*n^2-163350*n+205877)*a(n-4) +12*(6*n-35)*(6*n-31)*(3*n-16)*(2*n-11)*(3*n-17)*a(n-5) +a(n-6). - R. J. Mathar, Jan 28 2020
From Seiichi Manyama, Jan 20 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-6)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/6,n)/k!. (End)

A380310 Expansion of e.g.f. exp( 1 - 1/(1-5*x)^(1/5) ).

Original entry on oeis.org

1, -1, -5, -49, -719, -14077, -344909, -10152829, -349045535, -13727327833, -607873987637, -29931556660105, -1622308999459631, -95982568510668373, -6155361624644676989, -425321834949751148053, -31502433469012320013631, -2489898822489054343250737, -209178052238110675644666341

Offset: 0

Seiichi Manyama, Jan 20 2025

sign

Seiichi Manyama, Table of n, a(n) for n = 0..355

Cf. A145561, A380308, A380309.

Cf. A000587, A008548, A028575.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(1-1/(1-5*x)^(1/5))))

a(n) = Sum_{k=0..n} 5^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-5)^n * n! * Sum_{k>=0} (-1)^k * binomial(-k/5,n)/k!.
a(0) = 1; a(n) = -Sum_{k=1..n} A008548(k) * binomial(n-1,k-1) * a(n-k).

A380307 Expansion of e.g.f. exp( (1+5*x)^(1/5) - 1 ).

Original entry on oeis.org

1, 1, -3, 25, -335, 6177, -144947, 4128937, -138327615, 5327738497, -231899041475, 11255588133945, -602683483719503, 35288931375293857, -2242963870471014963, 153791777744471484745, -11314787069889491407103, 889087243145447511507969, -74312052321224600661026051

Offset: 0

Seiichi Manyama, Jan 20 2025

sign

Seiichi Manyama, Table of n, a(n) for n = 0..355

Cf. A000806, A380208.

Cf. A000110, A028575.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1+5*x)^(1/5)-1)))

a(n) = Sum_{k=0..n} 5^(n-k) * Stirling1(n,k) * Bell(k).
a(n) = (1/e) * 5^n * n! * Sum_{k>=0} binomial(k/5,n)/k!.
a(0) = 1; a(n) = Sum_{k=1..n} (Product_{j=0..k-1} (-5*j+1)) * binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

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

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

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A015735 Row sums of triangle A004747.

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A016036 Row sums of triangle A000369.

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A028844 Row sums of triangle A013988.

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A380310 Expansion of e.g.f. exp( 1 - 1/(1-5*x)^(1/5) ).

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A380307 Expansion of e.g.f. exp( (1+5*x)^(1/5) - 1 ).

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