A049223 -id:A049223 - 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

A025758 5th-order Vatalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 11, 171, 3056, 58916, 1191376, 24896436, 532911346, 11617952106, 256966100966, 5750337968926, 129926216608236, 2959472057112396, 67877180959091156, 1566072624078270516, 36319953436423545851

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

a(n), n >= 1, = row sums of triangle A049223.

Cf. A000108, A025756, A025757, A025759, A025760, A025761, A025762, A025763.

# Based on Tani Akinari's formula.
h := (n,j) -> ((-1)^n/(-4)^j)*binomial(j/5,n+1)*hypergeom([1,n+1-j/5],[n+2], 1025): a := n -> 2^8*5^(2*n+1)*add(h(n,j), j=1..4):
seq(round(evalf(a(n),64)),n=0..16); # Peter Luschny, Sep 21 2015

Table[SeriesCoefficient[5/(4 + (1 - 25*x)^(1/5)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)

a(n):=(256/205)*41^(-n)*sum(sum((-4)^(-k)*(-1025)^m*binomial(k/5,m),k,0,4),m,0,n);  /* Tani Akinari, Sep 16 2015  */

G.f.: 5 / (4+(1-25*x)^(1/5)).
a(n) = sum(m=1..n-1, 5^(n-m)*m/n * sum(k=1..n-m, binomial(n+k-1,n-1) * sum(i=0..k, binomial(k,i) * 2^(k-i) * sum(j=0..i, binomial(j,-3*i+n-m-k+2*j) * (-1)^(j-i)*5^(j-i)*(-2)^(3*i-n+m+k-j) * binomial(i,j)))))+1. - Vladimir Kruchinin, Feb 09 2011
Conjecture: 41*n*(n-1)*(n-2)*(n-3)*a(n) -3*(1367*n-4100)*(n-1)*(n-2)*(n-3)*a(n-1) +50*(n-2)*(n-3)*(3077*n^2-21533*n+38136)*a(n-2) -250*(n-3)*(10265*n^3-123135*n^2+494446*n-664572)*a(n-3) +1875*(8575*n^4-154250*n^3+1039765*n^2-3112730*n+3491808)*a(n-4) -625*(5*n-22)*(5*n-21)*(5*n-24)*(5*n-23)*a(n-5)=0. - R. J. Mathar, Jul 28 2014
a(n) = (-1)^(n+1) * 5^(2*n+1) * Sum_{k>=0} (-1/4)^(k+1) * binomial(k/5,n). - Seiichi Manyama, Aug 04 2024

A048966 A convolution triangle of numbers obtained from A025748.

Original entry on oeis.org

1, 3, 1, 15, 6, 1, 90, 39, 9, 1, 594, 270, 72, 12, 1, 4158, 1953, 567, 114, 15, 1, 30294, 14580, 4482, 1008, 165, 18, 1, 227205, 111456, 35721, 8667, 1620, 225, 21, 1, 1741905, 867834, 287199, 73656, 15075, 2430, 294, 24, 1, 13586859, 6857136, 2328183, 623106, 136323, 24354, 3465, 372, 27, 1

Offset: 1

Wolfdieter Lang

A generalization of the Catalan triangle A033184.

			Triangle begins:
     1;
     3,    1;
    15,    6,    1;
    90,   39,    9,    1;
   594,  270,   72,   12,    1;
  4158, 1953,  567,  114,   15,    1;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A034000, A049213, A049223, A049224. a(n, 1)= A025748(n), a(n, 1)= 3^(n-1)*2*A034000(n-1)/n!, n >= 2. Row sums = A025756.

a048966 n k = a048966_tabl !! (n-1) !! (k-1)
a048966_row n = a048966_tabl !! (n-1)
a048966_tabl = [1] : f 2 [1] where
   f x xs = ys : f (x + 1) ys where
     ys = map (flip div x) $ zipWith (+)
          (map (* 3) $ zipWith (*) (map (3 * (x - 1) -) [1..]) (xs ++ [0]))
          (zipWith (*) [1..] ([0] ++ xs))
-- Reinhard Zumkeller, Feb 19 2014

a[n_, m_] /; n >= m >= 1 := a[n, m] = 3*(3*(n-1) - m)*a[n-1, m]/n + m*a[n-1, m-1]/n; a[n_, m_] /; n < m := 0; a[n_, 0] = 0; a[1, 1] = 1; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Apr 26 2011, after given formula *)

a(n, m) = 3*(3*(n-1)-m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f. for m-th column: ((1-(1-9*x)^(1/3))/3)^m.

a(n,m) = m/n * sum(k=0..n-m, binomial(k,n-m-k) * 3^k*(-1)^(n-m-k) * binomial(n+k-1,n-1)). - Vladimir Kruchinin, Feb 08 2011

A049224 A convolution triangle of numbers obtained from A025751.

Original entry on oeis.org

1, 15, 1, 330, 30, 1, 8415, 885, 45, 1, 232254, 26730, 1665, 60, 1, 6735366, 825858, 58320, 2670, 75, 1, 202060980, 25992252, 2003562, 106560, 3900, 90, 1, 6213375135, 830282805, 68351283, 4038741, 174825, 5355, 105, 1, 194685754230

Offset: 1

Wolfdieter Lang

a(n,1) = A025751(n); a(n,1)= 6^(n-1)*5*A034787(n-1)/n!, n >= 2.

G.f. for m-th column: ((1-(1-36*x)^(1/6))/6)^m.

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

Cf. A048966, A049223. Row sums = A025759.

T(n,m):=(m*sum(binomial(-m+2*i-1,i-1)*2^(2*n-2*i)*sum(binomial(k,n-k-i)*3^(k+i-m)*(-1)^(n-k-i)*binomial(n+k-1,n-1),k,0,n-i),i,m,n))/n; /* Vladimir Kruchinin, Dec 21 2011 */

a(n, m) = 6*(6*(n-1)-m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f.: [(1-(1-36*x)^(1/6))/6]^m=sum(n>=m, T(n,m)*x^n), T(n,m)=(m*sum(i=m..n, binomial(-m+2*i-1,i-1)*2^(2*n-2*i)*sum(k=0..n-i, binomial(k,n-k-i)*3^(k+i-m)*(-1)^(n-k-i)*binomial(n+k-1,n-1))))/n. - Vladimir Kruchinin, Dec 21 2011

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

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Magma

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A025758 5th-order Vatalan numbers (generalization of Catalan numbers).

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Maxima

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A048966 A convolution triangle of numbers obtained from A025748.

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Haskell

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A049224 A convolution triangle of numbers obtained from A025751.

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Maxima

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