A078741 -id:A078741 - cp's OEIS frontend

A090219 Signed triangle used to compute column sequences of array A078741 ((3,3)-Stirling2).

1, -1, 4, 1, -8, 10, -1, 12, -30, 20, 1, -64, 600, -1600, 1225, -1, 80, -1000, 4000, -6125, 3136, 1, -96, 1500, -8000, 18375, -18816, 7056, -1, 448, -21000, 280000, -1500625, 3687936, -4148928, 1728000, 1, -512, 28000, -448000, 3001250, -9834496, 16595712, -13824000, 4492125, -1

Offset: 3

Wolfdieter Lang, Dec 01 2003

The formula for the column no. k sequence of array A078741 is c(k;n) = b(k-2)*sum(a(k,m)*fallfac(m+2,3)^n,m=1..k-2),n>=0, k>=3 and fallfac(p,3) and b(n) are defined in the formula below.

			The third (k=5) column sequence of array A078741 is A078741(n+3,5)=c(5; n)= b(3)*(1*(3*2*1)^n -8*(4*3*2)^n +10*(5*4*3)^n), with b(3)= N(3)/A090220(3)=3/1=3, n>=0. This is 9*A089518.
The fifth (k=7) column sequence of array A078741 is A078741(n+3,7)=c(7; n)= b(5)*(1*(3*2*1)^n -64*(4*3*2)^n +600*(5*4*3)^n -1600*(6*5*4)^n +1225*(7*6*5)^n), with b(5)= N(5)/A090220(5)=3/2, n>=0. This is the sequence [243, 149580, 49658508, 13062960720,... ] which has a factor of 27.
Triangle begins:
  [1];
  [-1,4];
  [1,-8,10];
  [-1,12,-30,20];
  [1,-64,600,-1600,1225];
  ...

Wolfdieter Lang, First 10 rows.

Companion sequence A090220 for denominators D(m).

a(n, m) = A089505(n-2, m)*(sum(A089517(n, p)/fallfac(m+2, 3)^p, p=0..floor(2*(n-3)/3)))/b(n-2), n>=3, 1<= m<= n-2, else 0; with fallfac(q, 3)=A008279(q, 3)=q*(q-1)*(q-2) and b(n)=N(n)/D(n) where D(n) := A090220(n) and N(n) is given in A090220 for n=1..26.

A089507 Second column of triangle A089504 and second column of array A078741 divided by 18.

Original entry on oeis.org

1, 30, 756, 18360, 441936, 10614240, 254788416, 6115201920, 146766525696, 3522406694400, 84537821131776, 2028908069959680, 48693795855814656, 1168651113600245760, 28047626804770062336, 673143043784666480640

Offset: 0

Wolfdieter Lang, Dec 01 2003

Convolution of A000400 (powers of 6) with A009968 (powers of 24).

Vincenzo Librandi, Table of n, a(n) for n = 0..730
Index entries for linear recurrences with constant coefficients, signature (30,-144).

Cf. A089505, A089513, A089514.

[6^n*(4^(n+1)-1)/3: n in [0..15]]; // Vincenzo Librandi, Oct 18 2017

CoefficientList[Series[1/((1-6x)(1-24x)),{x,0,20}],x] (* or *) LinearRecurrence[{30,-144},{1,30},20] (* Harvey P. Dale, Sep 25 2017 *)

G.f.: 1/((1-3*2*1*x)*(1-4*3*2*x)).
a(n) = A089504(n+2, 2), n>=0.
a(n) = (4*(4*3*2)^n - (3*2*1)^n)/3 = (2^n)*(2^(2*(n+1))-1)*3^(n-1).
a(n) = 6^n*(4^(n+1)-1)/3. - Vincenzo Librandi, Oct 18 2017

A089517 Array used for numerators of g.f.s for column sequences of array A078741 ((3,3)-Stirling2).

Original entry on oeis.org

1, 18, 9, 432, 1, 672, 14400, 243, 47520, 648000, 27, 36396, 3790800, 38102400, 1, 9765, 5115888, 354715200, 2844979200, 1107, 2499552, 757646784, 39182330880, 263363788800, 54, 546453, 592216272, 123294623040, 5089348454400

Offset: 3

Wolfdieter Lang, Dec 01 2003

The row length sequence for this array is A004396(n-2)=floor((2*n-3)/3), n>=3: [1,1,2,3,3,4,5,5,6,7,7,8,9,9,10,...].

The g.f. G(m,x) for the m-th column sequence (with leading zeros) of array A078741 is given there. The recurrence is G(m,x) = x*(3*fallfac(m-1,2)*G(m-1,x) + 3*(m-2)*G(m-2,x) + G(m-3,x))/(1-fallfac(m,3)*x), m>=4, with inputs G(1,x)=0=G(2,x) and G(3,x)=x/(1-(3*2*1)*x); where fallfac(n,m) := A008279(n,m) (falling factorials). Computed from the Blasiak et al. reference, eqs. (20) and (21) with r=3: recurrence for S_{3,3}(n,k).

			[1]; [18]; [9,423]; [1,672,14400]; [243,47520,648000]; ...
G(4,x)/(x^2) = 18/((1-3*2*1*x)*(1-4*3*2*x)). kmax(4)=0, hence P(4,x)=a(4,0)=18; x^2 from x^ceiling(4/3).

W. Lang, First 11 rows.

a(n, m) from: sum(a(n, m)*x^m, m=0..kmax(n)) = G(n, x)* product(1-fallfac(p, 3)*x, p=3..n)/x^ceiling(n/3), n>=3, with G(n, x) defined from the recurrence given above and kmax(n) := A004523(n-3)= floor(2*(n-3)/3) = A004396(n-3)-1.

A089518 Third column (k=5) of array A078741 ((3,3)-Stirling2) divided by 9.

Original entry on oeis.org

1, 138, 10476, 683208, 42315696, 2570768928, 155010407616, 9318969502848, 559578466388736, 33585275183251968, 2015370124337581056, 120928294183739148288, 7255843732407562776576, 435354129897768445943808

Offset: 0

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.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
Index entries for linear recurrences with constant coefficients, signature (90, -1944, 8640).

Cf. A089513 (third column of A089504), A089519, A090219.

G.f.: (1+48*x)/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)).

a(n)= (10*(5*4*3)^n - 8*(4*3*2)^n + (3*2*1)^n)/3 = b(n) + 48*b(n-1), with b(n) := A089513(n).

A089519 Fourth column (k=6) of array A078741 ((3,3)-Stirling2).

Original entry on oeis.org

1, 882, 186876, 28245672, 3762380016, 474431543712, 58322293189056, 7082435837377152, 854925864902090496, 102893307861680404992, 12365333752840511118336, 1484928368468173355231232

Offset: 0

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.

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

Cf. A089514, A089517-8, A090219.

G.f.: (1+672*x+14400*x^2)/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)*(1-6*5*4*x)).

a(n)= 20*(6*5*4)^n -30*(5*4*3)^n + 12*(4*3*2)^n - (3*2*1)^n = b(n) + 672*b(n-1) + 14400*b(n-2), with b(n) := A089514(n).

A090212 Alternating row sums of array A078741 ((3,3)-Stirling2).

Original entry on oeis.org

1, -4, 73, -3241, 223546, -10884061, -5437091357, 4560715140638, -2741631069546683, 1315509914960956853, -135771066929217673256, -969783690708328561039261, 1943740128890758048004419957, -2140191682145533094039398047820

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, A090211. A069223 (non-alternating sum, generalized Bell-numbers).

a[n_] := -Sum[(-1)^k FactorialPower[k, 3]^n/k!, {k, 2, Infinity}]*E; Array[a, 14] (* Jean-François Alcover, Sep 01 2016 *)

a(n) := sum( A078741(n, k)*(-1)^(k+1), k=3..3*n), n>=1. a(0) := -1 may be added.
a(n) = -sum(((-1)^k)*(fallfac(k, 3)^n)/k!, k=3..infinity)*exp(1), with fallfac(k, 3)=A008279(k, 3)=k*(k-1)*(k-2) and n>=1. This produces also a(0)=-1.
E.g.f. if a(0)=-1 is added: -exp(1)*(sum(((-1)^k)*exp(fallfac(k, 3)*x)/k!, k=3..infinity)+1/2). Similar to derivation on top of p. 4656 of the Schork reference.

A090220 Denominators used in A090219 to compute formula for column sequences of array A078741.

Original entry on oeis.org

1, 1, 1, 1, 2, 10, 20, 70, 560, 1680, 2800, 30800, 369600, 800800, 11211200, 168168000, 448448000, 7623616000, 137225088000, 434546112000, 8690922240000, 182509367040000, 669201012480000, 15391623287040000, 369398958888960000

Offset: 1

Wolfdieter Lang, Dec 01 2003

The corresponding numerator sequence is N(n) := [1, 6, 3, 1, 3, 3, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] for n=1..26.

			The fifth (k=7) column of A078741 needs in A090219 the factor b(5) := N(5)/a(5)= 3/2.

a(n) = lcm(seq(denominator(a(n+2, m))), m=1..n)), with the a(n, m) formula of A090219(n, m) but without the 1/b(n-2) factor and lcm denotes the least common multiple of a set of numbers.

N(n) := gcd(seq(numerator(a(n+2, m))), m=1..n)), with the a(n, m) formula of A090219(n, m) but without the 1/b(n-2) factor and gcd denotes the greatest common divisor > 1 of a set of numbers.

A078739 Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).

Original entry on oeis.org

1, 2, 4, 1, 4, 32, 38, 12, 1, 8, 208, 652, 576, 188, 24, 1, 16, 1280, 9080, 16944, 12052, 3840, 580, 40, 1, 32, 7744, 116656, 412800, 540080, 322848, 98292, 16000, 1390, 60, 1, 64, 46592, 1446368, 9196992, 20447056, 20453376, 10564304, 3047520, 511392, 50400

Offset: 1

N. J. A. Sloane, Dec 21 2002

A generalization of the Stirling2 numbers S_{1,1} from A008277.
The g.f. for column k=2*K is (x^K)*pe(K,x)*d(k,x) and for k=2*K+1 it is (x^K)*po(K,x)*2*(K+1)*K*d(k,x), K>= 1, with d(k,x) := 1/product(1-p*(p-1)*x,p=2..k) and the row polynomials pe(n,x) := sum(A089275(n,m)*x^m,m=0..n-1) and po(n,x) := sum(A089276(n,m)*x^m,m=0..n-1). - Wolfdieter Lang, Nov 07 2003
The formula for the k-th column sequence is given in A089511.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_2 (the disjoint union of n copies of the complete graph K_2). An example is given below. - Peter Bala, Aug 15 2013

			From _Peter Bala_, Aug 15 2013: (Start)
The table begins
n\k | 2    3    4    5    6   7   8
= = = = = = = = = = = = = = = = = =
  1 | 1
  2 | 2    4    1
  3 | 4   32   38   12    1
  4 | 8  208  652  576  188  24   1
...
Graph coloring interpretation of T(2,3) = 4: The graph 2K_2 is 2 copies of K_2, the complete graph on 2 vertices:
o---o  o---o
a   b  c   d
The four 3-colorings of 2K_2 are ac|b|d, ad|b|c, bc|a|d and bd|a|c. (End)

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, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
Steve Butler, Fan Chung, Jay Cummings, R. L. Graham, Juggling card sequences, arXiv:1504.01426 [math.CO], 2015.
Leonard Carlitz, On Arrays of Numbers, Am. J. Math., 54,4 (1932) 739-752. [Eqs. (3) and (4) with lambda = 0, mu = 2, a_{n,k-1} = a(n, k).- _Wolfdieter Lang_, Jan 30 2020 ]
P. Codara, O. M. D’Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers arXiv:1308.1700v1 [cs.DM]
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
S.-M. Ma, T. Mansour, M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

Row sums give A020556. Triangle S_{1, 1} = A008277, S_{2, 1} = A008297 (ignoring signs), S_{3, 1} = A035342, S_{3, 2} = A078740, S_{3, 3} = A078741. A090214 (S_{4,4}).
The column sequences are A000079(n-1)(powers of 2), 4*A016129(n-2), A089271, 12*A089272, A089273, etc.
Main diagonal is A217900.
Cf. A071951 (Legendre-Stirling triangle).
Cf. A122193, A055203.

# Note that the function implements the full triangle because it can be
# much better reused and referenced in this form.
A078739 := proc(n,k) local r;
add((-1)^(n-r)*binomial(n,r)*combinat[stirling2](n+r,k),r=0..n) end:
# Displays the truncated triangle from the definition:
seq(print(seq(A078739(n,k),k=2..2*n)),n=1..6); # Peter Luschny, Mar 25 2011

t[n_, k_] := Sum[(-1)^(n-r)*Binomial[n, r]*StirlingS2[n+r, k], {r, 0, n}]; Table[t[n, k], {n, 1, 7}, {k, 2, 2*n}] // Flatten (* Jean-François Alcover, Apr 11 2013, after Peter Luschny *)

a(n, k) = sum(binomial(k-2+p, p)*A008279(2, p)*a(n-1, k-2+p), p=0..2) if 2 <= k <= 2*n for n>=1, a(1, 2)=1; else 0. Here A008279(2, p) gives the third row (k=2) of the augmented falling factorial triangle: [1, 2, 2] for p=0, 1, 2. From eq.(21) with r=2 of the Blasiak et al. paper.
a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*A008279(p, 2)^n, p=2..k) for 2 <= k <= 2*n, n>=1. From eq.(19) with r=2 of the Blasiak et al. paper.
a(n, k) = sum(A071951(n, j)*A089503(j, 2*j-k+1), j=ceiling(k/2)..min(n, k-1)), 1<=n, 2<=k<=2n; relation to Legendre-Stirling triangle. Wolfdieter Lang, Dec 01 2003
a(n, k) = A122193(n,k)*2^n/k! - Peter Luschny, Mar 25 2011
E^n = sum_{k=2}^(2n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^2d^2/dx^2.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*sum {k = 0..inf} (k*(k-1))^n*x^k/k!. - Peter Bala, Aug 15 2013

More terms from Wolfdieter Lang, Nov 07 2003

A069223 Generalized Bell numbers: row 3 of A090210.

Original entry on oeis.org

1, 1, 34, 2971, 513559, 149670844, 66653198353, 42429389528215, 36788942253042556, 41888564490333642283, 60862147523250910055785, 110264570238241604072673394, 244397290937585028603794094349, 652229940568729289038518033117685, 2067551365133160531453420400711013314, 7694635622932764203876848262780670955447

Offset: 0

Karol A. Penson, Apr 12 2002

a(n) occurs in the process of normal ordering of the n-th power of a product of the cubes of the boson creation and boson annihilation operators.
a(11) = 110264570238241604072673394 =~ 10^26.
From Peter Luschny, Mar 27 2011: (Start)
Let B_{m}(x) = sum_{j>=0}(exp(j!/(j-m)!*x-1)/j!) then a(n) = n! [x^n] taylor(B_{3}(x)), where [x^n] denotes the coefficient of x^n in the Taylor series for B_{3}(x).
a(n) is row 3 of the square array representation of A090210. (End)

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, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Codara, O. M. D'Antona, and P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
P. Codara, O. M. D’Antona, and P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, Discrete Math. 318 (2014), 53--57. MR3141626
S.-M. Ma, T. Mansour, and M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).
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
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A000110 and A020556, if k+3 is replaced by k+1 or k+2, respectively.

Cf. A090210.

A069223 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(4,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*6^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A069223(n),n=1..15); # Peter Luschny, Mar 30 2011

f[n_] := f[n] = Sum[(k + 3)!^n/((k + 3)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 9}]
a[n_] := (* row sum of A078741 *) Sum[(-1)^k*Sum[(-1)^p*((p - 2)*(p - 1)*p)^n*Binomial[k, p], {p, 3, k}]/k!, {k, 3, 3n}]; Array[a, 15] (* Jean-François Alcover, Sep 01 2015 *)

default(realprecision, 500); for(n=0, 20, print1(if(n==0, 1, round(exp(-1)*suminf(k=0, ((k+3)!)^n/( (k+3)!*(k!)^n)))), ", ")) \\ G. C. Greubel, May 15 2018

a(n) = exp(-1) * Sum_{k>=0} ((k+3)!)^n/((k+3)!*(k!)^n), n>=1. This is a Dobinski-type summation formula.
a(n) = exp(-1) * Sum_{k>=3} (k*(k-1)*(k-2))^n/k!, n>=1. Usually a(0) := 1. (From eq.(26) with r=3 of the Schork reference; rewritten original eq.(25) with r=3 of the Blasiak et al. reference.)
E.g.f. with a(0) := 1: (sum((exp(k*(k-1)*(k-2)*x))/k!, k=3..infinity)+5/2)/exp(1). From top of p. 4656 with r=3 of the Schork reference.

Edited by Robert G. Wilson v, Apr 30 2002

a(0)=1 prepended by Alois P. Heinz, Aug 01 2016

A089504 A generalization of triangle A071951 (Legendre-Stirling).

Original entry on oeis.org

1, 6, 1, 36, 30, 1, 216, 756, 90, 1, 1296, 18360, 6156, 210, 1, 7776, 441936, 387720, 31356, 420, 1, 46656, 10614240, 23705136, 4150440, 119556, 756, 1, 279936, 254788416, 1432922400, 521757936, 29257200, 373572, 1260, 1, 1679616

Offset: 1

Wolfdieter Lang, Dec 01 2003

This triangle underlies the array entry A078741 ((3,3)-generalized Stirling2).

For the computation of the column sequences see A089505.

			[1]; [6,1]; [36,30,1]; [216,756,90,1]; ...
a(3,2) = 30 = ((-1)*(3*2*1)^1 + 4*(4*3*2)^1)/3.

R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.
W. Lang, First 8 rows.

Cf. A071951 (Legendre-Stirling, (2, 2) case).
The column sequences (without leading zeros) are A000400 (powers of 6), A089507, A089513-4, etc.
Cf. A008279, A089505, A089506.

max = 10; f[m_] := 1/Product[1 - FactorialPower[r + 2, 3]*x, {r, 1, m}]; col[m_] := CoefficientList[f[m] + O[x]^(max - m + 1), x]; a[n_, m_] := col[m][[n - m + 1]]; Table[a[n, m], {n, 1, max}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

G.f. for m-th column sequence (without leading zeros and m>=1) is 1/Product_{r=1..m} 1-fallfac(r+2, 3)*x with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m) = Sum_{p=1..m} A089505(m, p)*((p+2)*(p+1)*p)^(n-m))/D(m) if n>=m>=1 else 0; with D(m) := A089506(m).

cp's OEIS Frontend

A090219 Signed triangle used to compute column sequences of array A078741 ((3,3)-Stirling2).

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A089507 Second column of triangle A089504 and second column of array A078741 divided by 18.

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A089517 Array used for numerators of g.f.s for column sequences of array A078741 ((3,3)-Stirling2).

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A089518 Third column (k=5) of array A078741 ((3,3)-Stirling2) divided by 9.

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A089519 Fourth column (k=6) of array A078741 ((3,3)-Stirling2).

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A090212 Alternating row sums of array A078741 ((3,3)-Stirling2).

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A090220 Denominators used in A090219 to compute formula for column sequences of array A078741.

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A078739 Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).

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Maple

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A069223 Generalized Bell numbers: row 3 of A090210.

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