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A130191 Square of the Stirling2 matrix A048993. 9
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

G. C. Greubel, Rows n=0..100 of triangle, flattened

W. 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.

Cf. A000110, A039810, A048993.

T(2n,n) gives A321712.

Sequence in context: A269951 A176056 A298213 * A054651 A292323 A059720

Adjacent sequences:  A130188 A130189 A130190 * A130192 A130193 A130194

KEYWORD

nonn,tabl,easy

AUTHOR

Wolfdieter Lang, Jun 01 2007

STATUS

approved

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Last modified July 2 08:02 EDT 2020. Contains 335398 sequences. (Running on oeis4.)