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A130191 Square of the Stirling2 matrix A048993. 10
1, 0, 1, 0, 2, 1, 0, 5, 6, 1, 0, 15, 32, 12, 1, 0, 52, 175, 110, 20, 1, 0, 203, 1012, 945, 280, 30, 1, 0, 877, 6230, 8092, 3465, 595, 42, 1, 0, 4140, 40819, 70756, 40992, 10010, 1120, 56, 1, 0, 21147, 283944, 638423, 479976, 156072, 24570, 1932, 72, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Without row n=0 and column m=0 this is triangle A039810.
This is an associated Sheffer matrix with e.g.f. of the m-th column (exp(f(x)-1))^m)/m! with f(x)=:exp(x)-1.
The triangle is also called the exponential Riordan array [1, exp(exp(x)-1)]. - Peter Luschny, Apr 19 2015
Also the Bell transform of shifted Bell numbers A000110(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
LINKS
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First 10 rows and more
John Riordan, Letter, Apr 28 1976. (See third page)
FORMULA
a(n,m) = Sum_{k=m..n} S2(n,k) * S2(k,m), n>=m>=0.
E.g.f. row polynomials with argument x: exp(x*f(f(z))).
E.g.f. column m: ((exp(exp(x)-1)-1)^m)/m!.
EXAMPLE
Triangle starts:
[1]
[0, 1]
[0, 2, 1]
[0, 5, 6, 1]
[0,15,32,12,1]
MAPLE
# The function BellMatrix is defined in A264428.
BellMatrix(n -> combinat:-bell(n+1), 9); # Peter Luschny, Jan 27 2016
MATHEMATICA
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[BellB[# + 1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
a[n_, m_]:= Sum[StirlingS2[n, k]*StirlingS2[k, m], {k, m, n}]; Table[a[n, m], {n, 0, 100}, {m, 0, n}]//Flatten (* G. C. Greubel, Jul 10 2018 *)
PROG
(Sage) # uses[riordan_array from A256893]
riordan_array(1, exp(exp(x) - 1), 8, exp=true) # Peter Luschny, Apr 19 2015
(PARI) for(n=0, 10, for(m=0, n, print1(sum(k=m, n, stirling(n, k, 2)* stirling(k, m, 2)), ", "))) \\ G. C. Greubel, Jul 10 2018
CROSSREFS
Row sums: A000258. Alternating row sums: A130410.
T(2n,n) gives A321712.
Sequence in context: A269951 A176056 A298213 * A054651 A292323 A059720
KEYWORD
nonn,tabl,easy
AUTHOR
Wolfdieter Lang, Jun 01 2007
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)