A090217 -id:A090217 - cp's OEIS frontend

A090435 Triangle of signed numbers used for the computation of the column sequences of triangle A090217.

1, -1, 6, 1, -48, 147, -5, 1584, -24255, 50176, 1, -1980, 121275, -1003520, 1571724, -41, 496980, -113458275, 2950635520, -16174611684, 20412000000, 45182, -3322062810, 2744728561050, -206756932157440, 3081396966348393, -12443694076800000, 13160600037440625, -1294492177294

Offset: 1

Wolfdieter Lang, Dec 01 2003

A090217(n+m,m)= sum(a(m,p)*((p+4)*(p+3)*(p+2)*(p+1)*p)^n,p=1..m)/D(m) with D(m) := A090436(m); m=1,2,..., n>=0.

			[1]; [ -1,6]; [1,-48,147]; [ -5,1584,-24255,50176]; ...
A090217(2+3,3) = 9086400 = (1*(5*4*3*2*1)^2 - 48*(6*5*4*3*2)^2 + 147*(7*6*5*4*3)^2)/100.
a(3,2)= -48 = 100*(-1)*((6*5*4*3*2)^2)/((6*5*4*3*2-5*4*3*2*1)*(7*6*5*4*3-6*5*4*3*2)).

W. Lang, First 8 rows.

a(n, m)= D(n)*((-1)^(n-m))*(fallfac(m+4, 5)^(n-1))/(product(fallfac(m+4, 5)-fallfac(r+4, 5), r=1..m-1)*product(fallfac(r+4, 5)-fallfac(m+4, 5), r=m+1..n)), with D(n) := A090436(n) and fallfac(n, m) := A008279(n, m) (falling factorials), 1<=m<=n else 0. (Replace in the denominator the first product by 1 if m=1 and the second one by 1 if m=n.)

A090436 Denominators used to compute the column sequences of triangle A090217.

Original entry on oeis.org

1, 5, 100, 27500, 687500, 7075062500, 3594287401375000, 81456034414253835114625000, 35683959264371194185124824344091625000, 140789696830026154896057937831705061406034875000

Offset: 1

Wolfdieter Lang, Dec 01 2003

a(n) = lcm(seq(denominator(a(n,m)),m=1..n)) with the a(n,m) formula given in A090435(n,m) but without the D(n) factor in front and lcm denotes the least common multiple of a set of numbers.

A090216 Generalized Stirling2 array S_{5,5}(n,k).

Original entry on oeis.org

1, 120, 600, 600, 200, 25, 1, 14400, 504000, 2664000, 4608000, 3501000, 1350360, 284800, 33800, 2225, 75, 1, 1728000, 371520000, 7629120000, 42762240000, 97388280000, 110386900800, 70137648000, 26920728000, 6548346000, 1039382000

Offset: 1

Wolfdieter Lang, Dec 01 2003

The row length sequence for this array is [1, 6, 11, 16, 21, 26, 31,...]= A016861(n-1), n>=1.

The g.f. for the k-th column, (with leading zeros and k>=5) is G(k,x)= x^ceiling(k/5)*P(k,x)/product(1-fallfac(p,5)*x,p=5..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A090222(k,m)*x^m,m=0..kmax(k)), k>=5, with kmax(k) := floor(4*(k-5)/5)= A090223(k-5). For the recurrence of the G(k,x) see A090222.

			Triangle begins:
  [1];
  [120,600,600,200,25,1];
  [14400,504000,2664000,4608000,3501000,1350360,284800,33800,2225,75,1];
  ...

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004; Phys. Lett. A 309 (3-4) (2003) 198-205.
Wolfdieter Lang, First 5 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A090217, A090209 (row sums), A090218 (alternating row sums).

fallfac[n_, k_] := Pochhammer[n-k+1, k]; a[n_, k_] := (((-1)^k)/k!)*Sum[((-1)^p)*Binomial[k, p]*fallfac[p, 5]^n, {p, 5, k}]; Table[a[n, k], {n, 1, 5}, {k, 5, 5*n}] // Flatten (* Jean-François Alcover, Mar 05 2014 *)

from sympy import binomial, factorial, ff
def a(n, k): return sum((-1)**p * binomial(k, p) * ff(p, 5)**n for p in range(5, k+1)) * (-1)**k / factorial(k) # David Radcliffe, Jul 01 2025

a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*fallfac(p, 5)^n, p=5..k), with fallfac(p, 5) := A008279(p, 5)=product(p+1-q, q=1..5); 5<= k <= 5*n, n>=1, else 0. From eq.(19) with r=5 of the Blasiak et al. reference.

E^n = Sum_{k=5..5n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^5d^5/dx^5.

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A090435 Triangle of signed numbers used for the computation of the column sequences of triangle A090217.

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A090436 Denominators used to compute the column sequences of triangle A090217.

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A090216 Generalized Stirling2 array S_{5,5}(n,k).

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