A078739 -id:A078739 - cp's OEIS frontend

A089275 Coefficient triangle of polynomials used for numerator of g.f.s for column sequences of array A078739.

1, 1, 18, 1, 118, 600, 1, 412, 11772, 35280, 1, 1060, 97308, 1494576, 3265920, 1, 2270, 508708, 23753736, 249815520, 439084800, 1, 4298, 1989148, 218417400, 6710001408, 54187574400, 80951270400, 1, 7448, 6355048, 1402502400

Offset: 1

Wolfdieter Lang, Nov 07 2003

The polynomials are pe(n,x) := sum(a(n,m)*x^m,m=0..n-1). Companion polynomials are po(n,x) := sum(b(n,m)*x^m,m=0..n-1) with b(n,m) := A089276(n,m).

Wolfdieter Lang, First 7 rows, also for A089276

Cf. A078739, A089276.

Combined recursion for polynomials pe(n, x) and po(n, x) defined above: pe(n, x)= 4*(2*n-1)*n*(n-1)*x*po(n-1, x) + (1-(2*n-1)*(2*n-2)*x)*pe(n-1, x) and po(n, x) = 2*(pe(n, x) + ((n-1)/2)*(1-2*n*(2*n-1)*x)*po(n-1, x))/(n+1), n >= 2,  with po(1, x) = 1 = pe(1,x). (Corrected Wolfdieter Lang, Apr 11 2013)
Rewritten recursion for polynomial po: po(n, x) = (2*(1 - 2*(2*n-1)*(n-1)*x)*pe(n-1, x) + (n-1)*(1 + 6*n*(2*n-1)*x)* po(n-1, x))/(n+1), with pe(n,x) from above. - Wolfdieter Lang, Apr 11 2013
Combined recursion with b(n, m) := A089276(n, m): a(n, m) = a(n-1, m) - 2*(2*n-1)*(n-1)*a(n-1, m-1) + 4*n*(2*n-1)*(n-1)*b(n-1, m-1) and b(n, m) = (-2*n*(2*n-1)*(n-1)*b(n-1, m-1) + (n-1)*b(n-1, m) + 2*a(n, m))/(n+1), with n >= m+1 >= 2 and a(1, 0)= 1 = b(1, 0), else 0.
Rewritten recursion for triangle b: b(n, m) = (6*n*(2*n-1)*(n-1)*b(n-1, m-1) + (n-1)*b(n-1, m) + 2*a(n-1, m) - 4*(2*n-1)*(n-1)*a(n-1, m-1))/(n+1), with a(n, m) from above. - Wolfdieter Lang, Apr 11 2013

A089276 Coefficient triangle of polynomials used for numerator of g.f.s for column sequences of array A078739.

Original entry on oeis.org

1, 1, 8, 1, 48, 180, 1, 160, 3204, 8064, 1, 400, 24972, 311328, 604800, 1, 840, 125468, 4654656, 42454080, 68428800, 1, 1568, 476728, 40968960, 1073980368, 7803233280, 10897286400, 1, 2688, 1490328, 254542720, 15076235088, 306406471680

Offset: 1

Wolfdieter Lang, Nov 07 2003

The polynomials are po(n,x) := sum(a(n,m)*x^m,m=0..n-1). Companion polynomials are pe(n,x) := sum(b(n,m)*x^m,m=0..n-1) with b(n,m) := A089275(n,m), n>=1, po(1,x) = 1 = pe(1,x).

For the recursion relation of these polynomials and the triangles see A089275.

A089271 Third column (k=4) of array A078739(n,k) ((2,2)-generalized Stirling2).

Original entry on oeis.org

1, 38, 652, 9080, 116656, 1446368, 17636032, 213311360, 2569812736, 30898216448, 371141389312, 4455873443840, 53483541999616, 641880868118528, 7703040602324992, 92439308337643520, 1109288626710839296

Offset: 0

Wolfdieter Lang, Nov 07 2003

The numerator of the g.f. is the n=2 row polynomial of the triangle A089275.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Index entries for linear recurrences with constant coefficients, signature (20, -108, 144).

Cf. A089272, A071951 (Legendre-Stirling triangle).

[6*12^n-6*6^n+2^n: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011

Table[6*12^n -6*6^n +2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)
LinearRecurrence[{20,-108,144},{1,38,652},20] (* Harvey P. Dale, Oct 22 2024 *)

for(n=0,30, print1(6*12^n -6*6^n +2^n, ", ")) \\ G. C. Greubel, Feb 07 2018

G.f.: (1+18*x)/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)).

a(n) = 6*12^n - 6*6^n + 2^n = d(n) + 18*d(n-1), n>=1, a(0)=1, with d(n) := A016309(n) = A071951(n+3, 3) = (24*12^n-15*6^n+2^n)/10.

A089272 Fourth column (k=5) of array A078739(n,k) ((2,2)- generalized Stirling2) divided by 12.

Original entry on oeis.org

1, 48, 1412, 34400, 766416, 16296448, 337709632, 6896540160, 139644851456, 2813500878848, 56517475402752, 1133320271749120, 22702062218039296, 454469171469877248, 9094518828981174272, 181952003020274401280

Offset: 0

Wolfdieter Lang, Nov 07 2003

The numerator of the g.f. is the n=2 row polynomial of the triangle A089276.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003), 198-205.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Index entries for linear recurrences with constant coefficients, signature (40, -508, 2304, -2880).

Cf. A071952, A089271, A089273, A071951 (Legendre-Stirling triangle).

LinearRecurrence[{40, -508, 2304, -2880}, {1, 48, 1412, 34400}, 16] (* Jean-François Alcover, Feb 28 2020 *) (* Jean-François Alcover, Feb 28 2020 *)

G.f. (1+8*x)/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)).

a(n)= (3500*20^n - 3780*12^n + 945*6^n - 35*2^n)/630 = d(n) + 8*d(n-1), with d(n) := A071952(n+4)= (2500*20^n - 2268*12^n + 405*6^n - 7*2^n)/630, n>=1.

A089511 Triangle of integers used to compute column sequences of array A078739 ((2,2)-Stirling2).

Original entry on oeis.org

1, -1, 3, 1, -6, 6, -1, 27, -108, 100, 1, -36, 216, -400, 225, -1, 135, -2160, 10000, -16875, 9261, 1, -162, 3240, -20000, 50625, -55566, 21952, -1, 567, -27216, 350000, -1771875, 4084101, -4302592, 1679616, 1, -648, 36288, -560000, 3543750, -10890936, 17210368, -13436928

Offset: 2

Wolfdieter Lang, Dec 01 2003

The k-th column sequence (without leading zeros) of A078739 is for even k: sum(a(k,m)*((m+1)*m)^n,m=1..k-1)/D(k) and for odd k it is: ((k^2-1)/2)*sum(a(k,m)*((m+1)*m)^n,m=1..k-1)/D(k), where D(k) := A089512(k) and n>=0, k>=2.

			[1]; [ -1,3]; [1,-6,6]; [ -1,27,-108,100]; ...
a(2,1)=A089512(2)*A089275(1,0)*A089278(1,1)/A089500(1)=1*1*1/1=1;
a(3,2)=A089512(3)*A089276(1,0)*A089278(2,2)/A089500(2)=2*1*3/2=3.
a(4,3)=1*(1+18/(4*3))*24/10 =6; a(5,4)= 18*(1+8/(5*4))*2500/630=100.
k=2 column sequence of A078739 is (1*(2*1)^n)/1 = 2^n. k=3: 4*(-1*(2*1)^n + 3*(3*2)^n)/2 (see A016129).

W. Lang, First 10 rows.

a(n, m) triangle 2<=n, 1<= m <= (n-1), else 0, with a(2*k, m)= D(2*k)*sum(A089275(k, p)/((m+1)*m)^p, p=0..k-1)*A089278(2*k-1, m)/A089500(2*k-1) and a(2*k+1, m)= D(2*k+1)*sum(A089276(k, p)/((m+1)*m)^p, p=0..k-1)*A089278(2*k, m)/A089500(2*k), where D(n) := A089512(n).

A090211 Alternating row sums of array A078739 ((2,2)-Stirling2).

Original entry on oeis.org

1, -1, -1, 41, -375, -3001, 177063, -990543, -144800527, 3644593711, 214013895023, -12488200175463, -553322483517383, 61495192102867639, 2469939623420627543, -448608666325921194271, -19104207797417792353951, 4742067751530355028847327

Offset: 1

Wolfdieter Lang, Dec 01 2003

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

Cf. -A000587(n) from Stirling2 case A008277 with a(0) := -1. A020556 (non-alternating sum, generalized Bell-numbers).

a[n_] := Sum[(-1)^k FactorialPower[k, 2]^n/k!, {k, 2, Infinity}]*E; Array[a, 18] (* Jean-François Alcover, Sep 01 2016 *)

a(n) := sum( A078739(n, m)*(-1)^m, m=2..2*n), n>=1. a(0) := +1 may be added.
a(n) = sum(((-1)^k)*(fallfac(k, 2)^n)/k!, k=2..infinity)*exp(1), with fallfac(k, 2)=A008279(k, 2)=k*(k-1) and n>=1. This produces also a(0)=1.
E.g.f. if a(0)=1 is added: exp(1)*(sum(((-1)^k)*exp(k*(k-1)*x)/k!, k=2..infinity)). Similar to derivation on top p. 4656 of the Schork reference.

A089273 Fifth column (k=6) of array A078739(n,k) ((2,2)-generalized Stirling2).

Original entry on oeis.org

1, 188, 12052, 540080, 20447056, 706827968, 23178048832, 736079932160, 22912552596736, 704164858293248, 21462936995648512, 650674662791229440, 19656291799888777216, 592413643343696150528, 17826953303927872110592

Offset: 0

Wolfdieter Lang, Nov 07 2003

The numerator of the g.f. is the m=3 row polynomial of the triangle A089275.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003), 198-205.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Index entries for linear recurrences with constant coefficients, signature (70, -1708, 17544, -72000, 86400).

Cf. A089272, A071951 (Legendre-Stirling triangle).

a:= n-> (Matrix([[12052,188,1,0,0]]). Matrix(5, (i,j)-> if (i=j-1) then 1 elif j=1 then [70,-1708,17544, -72000,86400][i] else 0 fi)^n)[1,3]: seq(a(n), n=0..30);  # Alois P. Heinz, Aug 14 2008

LinearRecurrence[{70, -1708, 17544, -72000, 86400}, {1, 188, 12052, 540080, 20447056}, 15] (* Jean-François Alcover, Feb 28 2020 *)

G.f.: (1+118*x+ 600*x^2)/Product_{p=1..5} (1-(p+1)*p*x).

a(n) = (2^n - 36*6^n + 36*6*12^n - 400*20^n + 75*3*30^n)/6 = d(n) + 118*d(n-1) + 600*d(n-2), n>=2, with d(n) := A089274(n)= A071951(n+5, 5)= (16875*30^n - 20000*20^n + 6048*12^n - 405*6^n + 2*2^n)/2520.

A089512 Denominators used in the computation of the column sequences of array A078739 ((2,2)-Stirling2).

Original entry on oeis.org

1, 2, 1, 18, 6, 360, 90, 12600, 2520, 680400, 113400, 52390800, 7484400, 5448643200, 681080400, 735566832000, 81729648000, 125046361440000, 12504636144000, 26134689540960000, 2375880867360000, 6585941764321920000

Offset: 2

Wolfdieter Lang, Dec 01 2003

See A089511 for the column sequence computation of A078739.

a(n) = lcm(seq(denominator(b(n, m)), m=1..n-1)), n>=2, where b(n, m) is defined like A089511(n, m) but without the factor D(n) and lcm stands for the least common multiple of a set of numbers.

a(n) = (sqrt(2)*n!/(8*2^(n/2)))*((n^2+4n+2*sqrt(2)+3)-(n^2+4n-2*sqrt(2)+3)*(-1)^n) [offset 0]. - Paul Barry, Sep 04 2007

A217900 O.g.f.: Sum_{n>=0} n^n * (n+1)^(n-1) * exp(-n(n+1)x) * x^n / n!.

Original entry on oeis.org

1, 1, 4, 38, 576, 12052, 322848, 10564304, 408903680, 18288706544, 928575662400, 52780935007968, 3321208845997056, 229232635832433664, 17221699990084108288, 1399139700462119135232, 122235936429355565580288, 11428226675376971405577984, 1138551595285580854471388160

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:
1 = Sum_{n>=0} (n+1)^(n-1) * exp(-(n+1)*x) * x^n/n!.
More generally, if we define a(n) for fixed integers m, t, and s>=0, by:
(0) Sum_{n>=0} m * n^(s*n) * (n*t+m)^(n-1) * exp(-n^s*(n*t+m)*x) * x^n/n! = Sum_{n>=0} a(n)*x^n
then the coefficients a(n) are integral and may be expressed by:
(1) a(n) = 1/n! * Sum_{k=0..n} m*(-1)^(n-k)*binomial(n,k) * k^(s*n) * (k*t+m)^(n-1).
(2) a(n) = 1/n! * [x^n] Sum_{k>=0} m*k^(s*k)*(k*t+m)^(k-1)*x^k / (1 + k^s*(k*t+m)*x)^(k+1).
(3) a(n) = 1/t^((s-1)*n) * [x^(s*n)] 1 + m*x*(1+m*x)^(n-1) / Product_{k=1..n} (1-k*t*x).
(4) a(n) = 1/t^((s-1)*n) * [x^(s*n)] 1 + m*x*(1-m*x)^(s*n) / Product_{k=1..n} (1-(k*t+m)*x).

			O.g.f.: A(x) = 1 + x + 4*x^2 + 38*x^3 + 576*x^4 + 12052*x^5 + 322848*x^6 +...
where
A(x) = 1 + 1^1*2^0*x*exp(-1*2*x) + 2^2*3^1*exp(-2*3*x)*x^2/2! + 3^3*4^2*exp(-3*4*x)*x^3/3! + 4^4*5^3*exp(-4*5*x)*x^4/4! + 5^5*6^4*exp(-5*6*x)*x^5/5! +...
simplifies to a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..345

Cf. A078739, A217899, A217901, A217902, A217903, A217904, A217905, A217910, A217913, A218300.

a[n_] := 1/n!*Sum[(-1)^(n-k)*Binomial[n, k]*k^n*(k+1)^(n-1), {k, 0, n}]; a[0]=1; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 06 2013 *)

{a(n)=polcoeff(sum(m=0,n,m^m*(m+1)^(m-1)*x^m*exp(-m*(m+1)*x+x*O(x^n))/m!),n)}

{a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+1)^(k-1)*x^k/(1+k*(k+1)*x +x*O(x^n))^(k+1)), n)}

{a(n)=1/n!*sum(k=0,n, (-1)^(n-k)*binomial(n,k)*k^n*(k+1)^(n-1))}

{a(n)=polcoeff(1+x*(1+x)^(n-1)/prod(k=0, n, 1-k*x +x*O(x^n)), n)}

{a(n)=polcoeff(1+x*(1-x)^n/prod(k=0, n, 1-(k+1)*x +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n==0,1,sum(k=0,n-1, binomial(n-1,k) * Stirling2(2*n-k-1,n)))} \\ Paul D. Hanna, Nov 13 2012
/* PARI Programs for the General Case (START) ...................... */

{a(n,m=1,t=1,s=1)=polcoeff(sum(k=0, n, m*k^(s*k)*(t*k+m)^(k-1)*exp(-k^s*(t*k+m)*x+x*O(x^n))*x^k/k!), n)}

{a(n,m=1,t=1,s=1)=(1/n!)*polcoeff(sum(k=0, n, m*k^(s*k)*(t*k+m)^(k-1)*x^k/(1+k^s*(t*k+m)*x +x*O(x^n))^(k+1)), n)}

{a(n,m=1,t=1,s=1)=1/n!*sum(k=0, n, m*(-1)^(n-k)*binomial(n, k)*k^(s*n)*(t*k+m)^(n-1))}

{a(n,m=1,t=1,s=1)=(1/t^((s-1)*n))*polcoeff(1+m*x*(1+m*x)^(n-1)/prod(k=0, n, 1-t*k*x +x*O(x^(s*n))), s*n)}

{a(n,m=1,t=1,s=1)=(1/t^((s-1)*n))*polcoeff(1+m*x*(1-m*x)^(s*n)/prod(k=0, n, 1-(t*k+m)*x +x*O(x^(s*n))), s*n)}
/* (END) ........................................................... */

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+1)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+1)^(k-1)*x^k / (1 + k*(k+1)*x)^(k+1).
a(n) = [x^n] 1 + x*(1+x)^(n-1) / Product_{k=1..n} (1-k*x).
a(n) = [x^n] 1 + x*(1-x)^(n-1) / Product_{k=1..n} (1-(k+1)*x).
a(n) = A078739(n,n) for n>=1.
a(n) = Sum_{k=0..n-1} binomial(n-1,k) * Stirling2(2*n-k-1,n) for n>0, where Stirling2(n,k) = A008277(n,k). - Paul D. Hanna, Nov 13 2012
a(n) ~ 2^(2*n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... = 2*A106533. - Vaclav Kotesovec, May 09 2014

A020556 Number of oriented multigraphs on n labeled arcs (without loops).

Original entry on oeis.org

1, 1, 7, 87, 1657, 43833, 1515903, 65766991, 3473600465, 218310229201, 16035686850327, 1356791248984295, 130660110400259849, 14177605780945123273, 1718558016836289502159, 230999008481288064430879, 34208659263890939390952225, 5549763869122023099520756513

Offset: 0

Gilbert Labelle (gilbert(AT)lacim.uqam.ca) and Simon Plouffe

nonn

Generalized Bell numbers: a(n) = Sum_{k=2..2*n} A078739(n,k), n >= 1.
Let B_{m}(x) = Sum_{j>=0} exp(j!/(j-m)!*x-1)/j! then
a(n) = n! [x^n] taylor(B_{2}(x)), where [x^n] denotes the coefficient of x^n in the Taylor series for B_{2}(x). a(n) is row 2 of the square array representation of A090210. - Peter Luschny, Mar 27 2011
Also the number of set partitions of {1,2,...,2n+1} such that the block |n+1| is a part but no block |m| with m < n+1. - Peter Luschny, Apr 03 2011

			Example: For n = 2 the a(2) = 7 are the number of set partitions of 5 such that the block |3| is a part but no block |m| with m < 3: 3|1245, 3|4|125, 3|5|124, 3|12|45, 3|14|25, 3|15|24, 3|4|5|12. - _Peter Luschny_, Apr 05 2011

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Alois P. Heinz, Table of n, a(n) for n = 0..288
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Codara, O. M. D'Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
Peter Luschny, Set partitions
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
M. Riedel, Set partitions of unique elements from an n-by-m matrix where elements from the same row may not be in the same partition, Mathematics Stack Exchange.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A020554, A014500, A020558, A090210, A095149, A106436, A182930.

A020556 := proc(n) local k;
add((-1)^(n+k)*binomial(n,k)*combinat[bell](n+k),k=0..n) end:
seq(A020556(n),n=0..17); # Peter Luschny, Mar 27 2011
# Uses floating point arithmetic, increase working precision for large n.
A020556 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(3,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*2^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A020556(n),n=0..15); # Peter Luschny, Mar 30 2011
T := proc(n, k) option remember;
  if n = 1 then 1
elif n = k then T(n-1,1) - T(n-1,n-1)
else T(n-1,k) + T(n, k+1) fi end:
A020556 := n -> T(2*n+1,n+1);
seq(A020556(n), n = 0..99); # Peter Luschny, Apr 03 2011

f[n_] := f[n] = Sum[(k + 2)!^n/((k + 2)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 16}]
(* Second program: *)
a[n_] := Sum[(-1)^k*Binomial[n, k]*BellB[2n-k], {k, 0, n}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 11 2017, after Vladeta Jovovic *)

a(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(k=0, n, (-1)^k*binomial(n,k)*polcoef(bell, 2*n-k))} \\ Andrew Howroyd, Jan 13 2020

a(n) = e*Sum_{k>=0} ((k+2)!^n/(k+2)!)*(k!^n), n>=1.
a(n) = (1/e)*Sum_{k>=2} (k*(k-1))^n/k!, n >= 1. a(0) := 1. (From eq.(26) with r=2 of the Schork reference.)
E.g.f.: (1/e)*(2 + Sum_{k>=2} ((exp(k*(k-1)*x))/k!)) (from top of p. 4656 of the Schork reference).
a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*Bell(2*n-k). - Vladeta Jovovic, May 02 2004
a(n) = A095149(2n,n). - Alois P. Heinz, Dec 20 2018
a(n) = A106436(2n,n) = A182930(2n+1,n+1). - Alois P. Heinz, Jan 29 2019

Edited by Robert G. Wilson v, Apr 30 2002

cp's OEIS Frontend

A089275 Coefficient triangle of polynomials used for numerator of g.f.s for column sequences of array A078739.

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A089276 Coefficient triangle of polynomials used for numerator of g.f.s for column sequences of array A078739.

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A089271 Third column (k=4) of array A078739(n,k) ((2,2)-generalized Stirling2).

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A089272 Fourth column (k=5) of array A078739(n,k) ((2,2)- generalized Stirling2) divided by 12.

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A089511 Triangle of integers used to compute column sequences of array A078739 ((2,2)-Stirling2).

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A090211 Alternating row sums of array A078739 ((2,2)-Stirling2).

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A089273 Fifth column (k=6) of array A078739(n,k) ((2,2)-generalized Stirling2).

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Maple

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A089512 Denominators used in the computation of the column sequences of array A078739 ((2,2)-Stirling2).

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A217900 O.g.f.: Sum_{n>=0} n^n * (n+1)^(n-1) * exp(-n*(n+1)*x) * x^n / n!.

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A217900 O.g.f.: Sum_{n>=0} n^n * (n+1)^(n-1) * exp(-n(n+1)x) * x^n / n!.