A028844 -id:A028844 - cp's OEIS frontend

A013988 Triangle read by rows, the inverse Bell transform of n!*binomial(5,n) (without column 0).

1, 5, 1, 55, 15, 1, 935, 295, 30, 1, 21505, 7425, 925, 50, 1, 623645, 229405, 32400, 2225, 75, 1, 21827575, 8423415, 1298605, 103600, 4550, 105, 1, 894930575, 358764175, 59069010, 5235405, 271950, 8330, 140, 1, 42061737025, 17398082625, 3016869625, 289426830, 16929255, 621810, 14070, 180, 1

Offset: 1

Wolfdieter Lang

Previous name was: Triangle of numbers related to triangle A049224; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n, m) = S2p(-5; 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) = A008543(n-1).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			Triangle begins as:
          1;
          5,         1;
         55,        15,        1;
        935,       295,       30,       1;
      21505,      7425,      925,      50,      1;
     623645,    229405,    32400,    2225,     75,     1;
   21827575,   8423415,  1298605,  103600,   4550,   105,    1;
  894930575, 358764175, 59069010, 5235405, 271950,  8330,  140,   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.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
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, A008543, A049224, A264428.
Cf. A028844 (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), A011801 (m=5), this sequence (m=6).

function T(n,k) // T = A013988
  if k eq 0 then return 0;
  elif k eq n then return 1;
  else return (6*(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 *)
rows = 10;
b[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
A = Table[b[n, m], {n, 1, rows}, {m, 1, rows}] // Inverse // Abs;
A013988 = Table[A[[n, m]], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==0, 0, If[k==n, 1, (6*(n-1) -k)*T[n-1,k] +T[n-1, k-1]]];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 03 2023 *)

# 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(5, n), 8) # Peter Luschny, Jan 16 2016

T(n, m) = n!*A049224(n, m)/(m!*6^(n-m));

T(n+1, m) = (6*n-m)*T(n, m) + T(n, m-1), for n >= m >= 1, with T(n, m) = 0, n

E.g.f. of m-th column: ((1 - (1-6*x)^(1/6))^m)/m!.

Sum_{k=1..n} T(n, k) = A028844(n).

New name from Peter Luschny, Jan 16 2016

A157405 A partition product of Stirling_2 type [parameter k = 5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 55, 1, 105, 220, 935, 1, 425, 3300, 4675, 21505, 1, 3075, 47850, 84150, 129030, 623645, 1, 15855, 415800, 2323475, 2709630, 4365515, 415800, 2323475, 2709630, 4365515, 21827575, 1, 123515, 6394080, 51934575

Offset: 0

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144268.
Same partition product with length statistic is A013988.
Diagonal a(A000217) = A008543.
Row sum is A028844.

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

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404

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}(6*j - 1).

A028575 Row sums of triangle A011801.

Original entry on oeis.org

1, 5, 49, 721, 14177, 349141, 10334689, 357361985, 14137664833, 629779342213, 31195027543505, 1700812505769169, 101218448336028193, 6528869281965115541, 453720852957751220353, 33796334125623555379969, 2686138908337714715560577, 226908450494953996837748869

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.

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), this sequence (m=5), A028844 (m=6).

Cf. A011801.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-5*x)^(1/5)) - 1 ))); // G. C. Greubel, Oct 02 2023

Rest[With[{nn=20},CoefficientList[Series[Exp[1-(1-5x)^(1/5)]-1, {x,0,nn}], x] Range[0,nn]!]] (* Harvey P. Dale, Aug 02 2016 *)

def A028575_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-5*x)^(1/5)) -1 ).egf_to_ogf().list()
a=A028575_list(40); a[1:] # G. C. Greubel, Oct 02 2023

E.g.f.: exp(1 - (1-5*x)^(1/5)) - 1.
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)^4*d/dx. Cf. A001515, A015735 and A016036. - Peter Bala, Nov 25 2011
D-finite with recurrence: a(n) -20*(n-3)*a(n-1) +30*(5*n^2-35*n +62)*a(n-2) -100*(n-4)*(5*n^2-40*n+81)*a(n-3) +(5*n-22)*(5*n-21)*(5*n-24)*(5*n-23)*a(n-4) -a(n-5) = 0. - R. J. Mathar, Jan 28 2020
From Seiichi Manyama, Jan 20 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-5)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/5,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)

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A013988 Triangle read by rows, the inverse Bell transform of n!*binomial(5,n) (without column 0).

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A157405 A partition product of Stirling_2 type [parameter k = 5] with biggest-part statistic (triangle read by rows).

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A028575 Row sums of triangle A011801.

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A016036 Row sums of triangle A000369.

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