A006472 -id:A006472 - cp's OEIS frontend

A317059 a(n) is the number of time-dependent assembly trees satisfying the edge gluing rule for a complete graph on n vertices.

1, 1, 3, 21, 255, 4815, 130095, 4763115, 226955925, 13646570175, 1010560060125, 90363456777825, 9599238270346725, 1194935000536101825, 172283712268118826375, 28481473075454845070625, 5351643310498951112521875, 1134140509146174954631081875, 269235074280949277622074328375

Offset: 1

Nick Mayers, Robert Short, Aria Dougherty, Jul 20 2018

A time-dependent assembly tree for a connected graph G=(V, E) on n vertices is a rooted tree, each node of which is label a subset U of V and a nonnegative integer i such that:
1) each internal node has at least two children,
2) there are leaves labeled (v, 0) for each vertex v in V,
3) the label on the root is (V, m) for 1 <= m <= n-1,
4) for each node (U, i) with i5) if (U, i) and (U', i') are adjacent nodes with U a subset of U', then i6) for each 0 <= i <= m, there exists a node (U, i) with U a subset of V.
A time-dependent assembly tree is said to satisfy the edge gluing rule if each internal vertex v of G has exactly two children and if U_1 and U_2 are the labels of the children of internal vertex v, then there is an edge (v_1,v_2) in the edge set of G such that v_1 is in U_1 and v_2 is in U_2.
a(n) is also the number of labeled histories possible for n leaves if simultaneous bifurcations are allowed. a(n) is also the number of single-elimination sports tournament schedules possible for n teams if matches involve pairs of teams, arbitrarily many arenas are available, and labeled teams have been specified, but the bracket of matches has not been specified. - Noah A Rosenberg, Feb 20 2025

Seiichi Manyama, Table of n, a(n) for n = 1..261
M. Bona and A. Vince, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012.
Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307.
A. Dougherty, N. Mayers, and R. Short, How to Build a Graph in n Days: Some Variants on Graph Assembly, arXiv preprint arXiv:1807.08079 [math.CO], 2018.

Cf. A006472, A047781, A317057, A317058, A317060.

Nest[Function[{a, n}, Append[a, Sum[(n!/((2^j) j! (n - 2 j)!)) a[[n - j]], {j, Floor[n/2]}]]][#, Length@ # + 1] &, {1, 1}, 17] (* Michael De Vlieger, Jul 26 2018 *)

lista(nn) = my(v = vector(nn)); for (n=1, nn, if (n<=2, v[n] = 1, v[n] = sum(j=1, n\2, (n!/((2^j)*j!*(n-2*j)!))*v[n-j]))); v; \\ Michel Marcus, Aug 08 2018

@cached_function
def TimeDepenEdgeComp(n):
    if n==1:
        return 1
    elif n==2:
        return 1
    else:
        return sum((factorial(n)/((2^j)*factorial(j)*factorial(n-2*j)))*TimeDepenEdgeComp(n-j) for j in range(1, n//2+1))
print(",".join(str(TimeDepenEdgeComp(i)) for i in range(1, 20)))

a(n) = Sum_{j=1..floor(n/2)}(n!/((2^j)j!(n-2j)!))*a(n-j), a(1)=a(2)=1.

A330728 Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 7, 5, 5, 11, 16, 16, 27, 18, 61, 62, 272, 45, 123, 61, 1385, 105, 152, 272, 501, 211, 7936, 362

Offset: 1

Gus Wiseman, Dec 30 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(n) multisystems for n = 3, 6, 8, 9, 10, 12 (commas and outer brackets elided):
  11  {1}{12}  {1}{23}  {{1}}{{1}{22}}  {{1}}{{1}{12}}  {{1}}{{1}{23}}
      {2}{11}  {2}{13}  {{11}}{{2}{2}}  {{11}}{{1}{2}}  {{11}}{{2}{3}}
               {3}{12}  {{1}}{{2}{12}}  {{1}}{{2}{11}}  {{1}}{{2}{13}}
                        {{12}}{{1}{2}}  {{12}}{{1}{1}}  {{12}}{{1}{3}}
                        {{2}}{{1}{12}}  {{2}}{{1}{11}}  {{1}}{{3}{12}}
                        {{2}}{{2}{11}}                  {{13}}{{1}{2}}
                        {{22}}{{1}{1}}                  {{2}}{{1}{13}}
                                                        {{2}}{{3}{11}}
                                                        {{23}}{{1}{1}}
                                                        {{3}}{{1}{12}}
                                                        {{3}}{{2}{11}}

The version with distinct atoms is A006472.
The non-maximal version is A318846.
A tree version is A318848, with orderless version A318849.
The unlabeled version is A330664.
Final terms in each row of A330727.
See also A330675 (strongly normal), A330676 (normal), and A330726 (partition).
Cf. A000111, A001055, A005121, A292504, A292505, A317145, A330665, A330666.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[Reverse[FactorInteger[n]],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

a(2^n) = A006472(n).

a(prime(n)) = A000111(n - 1).

A331956 Triangle T(n,k) read by rows: number of rooted chains of length k in set partitions of n labeled points.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 14, 31, 18, 0, 1, 51, 255, 385, 180, 0, 1, 202, 2066, 6110, 6945, 2700, 0, 1, 876, 17549, 90839, 188510, 171045, 56700, 0, 1, 4139, 159615, 1364307, 4603620, 7314650, 5507460, 1587600

Offset: 0

S. R. Kannan, Rajesh Kumar Mohapatra, Feb 02 2020

Also the number of chains of length k in unordered set partitions of {1,2,...,n} such that the first term of the chains is either {{1}, {2},...,{n}} or {{1,2,..,n}}.

Number of rooted k-level fuzzy equivalence matrices of order n.

			Triangle T(n,k) begins:
n\k | 0 1   2     3     4      5      6     7
----+-----------------------------------------
  0 | 1
  1 | 0 1
  2 | 0 1   1
  3 | 0 1   4     3
  4 | 0 1  14    31    18
  5 | 0 1  51   255   385    180
  6 | 0 1 202  2066  6110   6945   2700
  7 | 0 1 876 17549 90839 188510 171045 56700
  ...
The T(3,2) = 4 in the lattice of set partitions of {1,2,3}:
{{1},{2},{3}} < {{1,2},{3}},
{{1},{2},{3}} < {{1,3},{2}},
{{1},{2},{3}} < {{1},{2,3}},
{{1},{2},{3}} < {{1,2,3}}.
Or,
{{1,2,3}} > {{1,2},{3}},
{{1,2,3}} > {{1,3},{2}},
{{1,2,3}} > {{1},{2,3}},
{{1,2,3}} > {{1},{2},{3}}.

Alois P. Heinz, Rows n = 0..140, flattened
S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
V. Murali, Equivalent finite fuzzy sets and Stirling numbers, Inf. Sci., 174 (2005), 251-263.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.

Cf. A000007 (column k=0), A057427 (column k=1), A058692 (column k=2), A006472 (diagonal), A331957 (row sums).

Cf. A000110, A008277, A048993, A328044, A330301, A330302.

b:= proc(n, k, t) option remember; `if`(k<0 or k>n, 0, `if`(k=1 or
      {n, k}={0}, 1, add(b(v, k-1, 1)*Stirling2(n, v), v=k..n-t)))
    end:
T:= (n, k)-> b(n, k, 0):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 09 2020

b[n_, k_, t_] := b[n, k, t] = If[k < 0 || k > n, 0, If[k == 1 || Union@{n, k} == {0}, 1, Sum[b[v, k - 1, 1]*StirlingS2[n, v], {v, k, n - t}]]];
T[n_, k_] := b[n, k, 0];
Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten

b(n, k, t) = {if (k < 0, return(0)); if ((n==0) && (k==0), return (1)); if ((k==1) && (n>0), return(1)); sum(v = k, n - t, if (k==1, 1, b(v, k-1, 1))*stirling(n, v, 2));}
T(n, k) = b(n, k, 0);
matrix(8,8,n, k, T(n-1, k-1)) \\ to see the triangle \\ Michel Marcus, Feb 09 2020

T(0, 0) = 1, T(0, k) = 0 for k > 0 and T(n, 1) = 1 for n > 1.

T(n, k) = Sum_{i_(k-1)=k-1..n-1} (Sum_{i_(k-2)=k-2..i_(k-1) - 1} (... (Sum_{i_2=2..i_3 - 1} (Sum_{i_1=1..i_2 - 1} Stirling2(n,i_(k-1)) * Stirling2(i_(k-1),i_(k-2)) * ... * Stirling2(i_3,i_2) * Stirling2(i_2,i_1)))...)), where 2 <= k <= n.

A155926 G.f. satisfies: A(x) = B(xA(x)) where A(x) = Sum_{n>=0} a(n)x^n/[n!(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!(n+1)!/2^n].

Original entry on oeis.org

1, 1, 4, 37, 621, 16526, 640207, 34039027, 2379382609, 211619306134, 23337543447296, 3125553148981176, 499716551101393705, 94016487294245251308, 20561796731966531616954, 5172827581575899147920471

Offset: 0

Paul D. Hanna, Jan 30 2009

nonn

			G.f.: A(x) = 1 + x + 4*x^2/3 + 37*x^3/18 + 621*x^4/180 + 16526*x^5/2700 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
B(x) = 1 + x + 1/3*x^2 + 1/18*x^3 + 1/180*x^4 +...+ x^n/[n!*(n+1)!/2^n] +... where
A(x) = B(x*A(x)) and B(x) = A(x/B(x)) ;
1/B(x) = 1 - x + 2*x^2/3 - 7*x^3/18 + 39*x^4/180 - 321*x^5/2700 +...+ (-1)^n*A103365(n)*x^n/[n!*(n+1)!/2^n] +...
Also, A(x) = C(x*A(x)^2) where:
C(x) = 1 + x - 2*x^2/3 + 19*x^3/18 - 379*x^4/180 + 12726*x^5/2700 +...+ A155927(n)*x^(n+1)/[n!*(n+1)!/2^n] +...
A(x)^2 = 1 + 2*x + 11*x^2/3 + 122*x^3/18 + 2302*x^4/180 + 66482*x^5/2700 +...

Cf. A105558, A105556, A001263, A103365, A006472, A090181, A155927, A155928 (A^2).

{a(n)=local(F=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(serreverse(x/F)/x,n)*n!*(n+1)!/2^n}

{a(n)=local(N=matrix(n+1, n+1, m, j, if(m>=j, binomial(m-1, j-1)*binomial(m, j-1)/j))); sum(j=0, n, (N^n)[n+1, j+1])/(n+1)}

a(n) = A105558(n)/(n+1) = A105556(2n,n)/(n+1) = [N^(n+1)](n+1,1)/(n+1) for n>=0, where N^(n+1) is the (n+1)-th matrix power of the Narayana triangle N=A001263.
G.f.: A(x) = Series_Reversion[x/B(x)]/x where B(x) = A(x/B(x)) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n].
G.f. satisfies: A(x) = C(x*A(x)^2) and C(x) = A(x/C(x)^2) where C(x) is the g.f. of A155927.

A259459 From higher-order arithmetic progressions.

Original entry on oeis.org

1, 18, 360, 9000, 283500, 11113200, 533433600, 30862944000, 2121827400000, 171160743600000, 16020645600960000, 1722947613266880000, 211061082625192800000, 29223842209642080000000, 4542220046298654720000000, 787620956028186728448000000

Offset: 0

N. J. A. Sloane, Jun 30 2015

nonn

The expression "2 over n!" in the article is A006472(n+1). It is used in A259459 - A378234 (C_1 - C_3) on page 13. - Georg Fischer, Dec 06 2024

Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Cf. A006472, A259460, A378234.

rV := proc(n,a,d)
        n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
end proc:
A259459 := proc(n)
        mul(rV(i,a,d),i=1..n+1) ;
        coeftayl(%,d=0,1) ;
        coeftayl(%,a=0,n) ;
end proc:
seq(A259459(n),n=1..15) ; # R. J. Mathar, Jul 14 2015

rV[n_, a_, d_] := n(n+1)/2*a + (n-1)n(n+1)/6*d;
A259459[n_] :=
   Product[rV[i, a, d], {i, 1, n+2}] //
   SeriesCoefficient[#, {d, 0, 1}]& //
   SeriesCoefficient[#, {a, 0, n+1}]&;
Table[A259459[n], {n, 0, 14}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

-2*n*a(n) +(n+3)*(n+2)^2*a(n-1)=0. - R. J. Mathar, Jul 15 2015
Conjectured g.f.: 3F0(4,3,3;;x/2). - R. J. Mathar, Aug 09 2015
a(n) = (n+3)!*(n+2)!/2^(n+2)*(n+1)*(n+2)/6. - Georg Fischer, Dec 06 2024

A128813 Triangle of coefficients of (x+1)(x+3)(x+6)...(x+n(n+1)/2).

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 10, 27, 18, 1, 20, 127, 288, 180, 1, 35, 427, 2193, 4500, 2700, 1, 56, 1162, 11160, 50553, 97200, 56700, 1, 84, 2730, 43696, 363033, 1512684, 2778300, 1587600, 1, 120, 5754, 141976, 1936089, 14581872, 57234924, 101606400, 57153600

Offset: 0

Miklos Kristof, Apr 10 2007

			(x+1)(x+3)(x+6)=x^3+10x^2+27x+18, so a(3,j)=1, 10, 27, 18.
The triangle begins:
  1
  1, 1
  1, 4,  3
  1, 10, 27, 18
  1, 20, 127, 288, 180
  1, 35, 427, 2193, 4500, 2700
  1, 56, 1162, 11160, 50553, 97200, 56700

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A006472 (right diagonal), A128814 (row sums).

for n from 1 to 9 do b[n]:=n*(n+1)/2 od: a[0,0]:=1:a[1,0]:=1:a[1,1]:=1:for i from 2 to 9 do a[i,0]:=1:for j from 1 to i-1 do a[i,j]:=b[i]*a[i-1,j-1]+ a[i-1,j] od:a[i,i]:=b[i]*a[i-1,i-1] od: seq(seq(a[i,j],j=0..i),i=0..9);

Flatten[Table[Reverse[CoefficientList[Expand[Times@@Table[x+(n(n+1))/2,{n,i}]],x]],{i,0,9}]] (* Harvey P. Dale, Nov 11 2011 *)

row(n) = Vec(prod(i=1, n, (x+i*(i+1)/2))); \\ Michel Marcus, Mar 18 2023

a(0,0)=1, a(1,0)=1, a(1,1)=1, a(i,j)=i*(i+1)/2*a(i-1,j-1)+a(i-1,j), j=0..i-1.
a(i,i) = i*(i+1)/2*a(i-1,i-1).
a(n,n) = Product_{k=1..n} k*(k+1)/2 = A006472(n+1)
Sum_{m=0..n} a(n,m) = Product_{k=1..n} k*(k+1)/2+1 = A128814(n).

A132818 The matrix product A127773 * A001263 of infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 3, 6, 18, 6, 10, 60, 60, 10, 15, 150, 300, 150, 15, 21, 315, 1050, 1050, 315, 21, 28, 588, 2940, 4900, 2940, 588, 28, 36, 1008, 7056, 17640, 17640, 7056, 1008, 36, 45, 1620, 15120, 52920, 79380, 52920, 15120, 1620, 45, 55, 2475, 29700, 138600, 291060

Offset: 1

Gary W. Adamson, Sep 02 2007

			First few rows of the triangle are:
1;
3, 3;
6, 18, 6;
10, 60, 60, 10;
15, 150, 300, 150, 15;
21, 315, 1050, 1050, 315, 21;
...

Cf. A127773, A001263, A002457 (row sums), A006472. A002695, A008459.

A132818 := proc(n,k)
    (n+1-k)*binomial(n+1,k)*binomial(n,k-1)/2 ;
end proc: # R. J. Mathar, Jul 29 2015

T(n,k) = A000217(n) * A001263(n,k).
Let a(n) = A006472(n), the 'triangular' factorial numbers. Then a(n)/(a(k)*a(n-k)) produces the present triangle (with a different offset). - Peter Bala, Dec 07 2011
T(n,k) = 1/2*(n+1-k)*C(n+1,k)*C(n,k-1), for n,k >= 1. O.g.f.: x*y/((1-x-x*y)^2 - 4*x^2*y)^(3/2) = x*y + x^2*(3*y + 3*y^2) + x^3*(6*y + 18*y^2 + 6*y^3) + .... Cf. A008459 with o.g.f.: x*y/((1-x-x*y)^2 - 4*x^2*y)^(1/2). Sum {k = 1..n-1} T(n,k)*2^(n-k) = A002695(n). - Peter Bala, Apr 10 2012

Corrected by R. J. Mathar, Jul 29 2015

A184358 a(n) = (n+1)!^2/2^n.

Original entry on oeis.org

1, 2, 9, 72, 900, 16200, 396900, 12700800, 514382400, 25719120000, 1556006760000, 112032486720000, 9466745127840000, 927741022528320000, 104370865034436000000, 13359470724407808000000, 1930443519676928256000000, 312731850187662377472000000

Offset: 0

Paul D. Hanna, Jan 16 2011

nonn

Self-convolution of A184359.

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 72*x^3 + 900*x^4 + 16200*x^5 +...
A(x)^(1/2) = 1 + x + 4*x^2 + 32*x^3 + 410*x^4 + 7562*x^5 + 188736*x^6 +...+ A184359(n)*x^n +...

Cf. A184359, A184360, A184361, A006472.

a[n_] := (n + 1)!^2/2^n; Array[a, 20, 0] (* Amiram Eldar, Jun 25 2022 *)

PARI
```
{a(n)=(n+1)!^2/2^n}
```

From Amiram Eldar, Jun 25 2022: (Start)
Sum_{n>=0} 1/a(n) = (BesselI(0, 2*sqrt(2)) - 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (1 - BesselJ(0, 2*sqrt(2)))/2. (End)

A259460 From higher-order arithmetic progressions.

Original entry on oeis.org

4, 220, 10500, 535500, 30870000, 2044828800, 156029328000, 13673998800000, 1369285948800000, 155756276676000000, 20005336176265440000, 2884501462544301600000, 464334381775424160000000, 83021688624014300160000000, 16408769917253890176000000000, 3569104362241728159962112000000, 850861011640079911341911040000000

Offset: 0

N. J. A. Sloane, Jun 30 2015

nonn

The expression "2 over n!" in the article is A006472(n+1). It is used in C_1, C_2, C_3 (A259459, A259460, A378234) on page 13. A_2 is A000914. - Georg Fischer, Dec 06 2024

Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Cf. A000914, A006472, A259459, A378234.

rV := proc(n,a,d)
    n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
end proc:
A259460 := proc(n)
    mul(rV(i,a,d),i=1..n+2) ;
    coeftayl(%,d=0,2) ;
    coeftayl(%,a=0,n) ;
end proc:
seq(A259460(n),n=1..18) ; # R. J. Mathar, Jul 14 2015

rV[n_, a_, d_] := n (n + 1)/2*a + (n - 1) n (n + 1)/6*d;
A259460[n_] :=
   Product[rV[i, a, d], {i, 1, n + 3}] //
   SeriesCoefficient[#, {d, 0, 2}] & //
   SeriesCoefficient[#, {a, 0, n + 1}] & ;
Table[A259460[n], {n, 0, 16}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

From Georg Fischer, Dec 06 2024: (Start)
a(n) = (n+4)!*(n+3)!/2^(n+3)/9 * (n+2)*(n+1)*(n+3)*(3*n+8)/24.
D-finite with recurrence: 2*n*(3*n+5)*a(n) - (n+3)^2*(n+4)*(3*n+8)*a(n-1) = 0. (End)

Typos in data corrected by Jean-François Alcover, Apr 27 2023

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cp's OEIS Frontend

A175553 Product of first k triangular numbers divided by the sum of first k triangular numbers is an integer.

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A317059 a(n) is the number of time-dependent assembly trees satisfying the edge gluing rule for a complete graph on n vertices.

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A330728 Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.

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A331956 Triangle T(n,k) read by rows: number of rooted chains of length k in set partitions of n labeled points.

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A155926 G.f. satisfies: A(x) = B(x*A(x)) where A(x) = Sum_{n>=0} a(n)*x^n/[n!*(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n].

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A259459 From higher-order arithmetic progressions.

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A128813 Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).

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A155926 G.f. satisfies: A(x) = B(xA(x)) where A(x) = Sum_{n>=0} a(n)x^n/[n!(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!(n+1)!/2^n].

A128813 Triangle of coefficients of (x+1)(x+3)(x+6)...(x+n(n+1)/2).