1,2

Row sums of A098546 give sequence A098545 and row sums of A036040 give sequence A000110 (the Bell numbers)

Equals column zero of triangle A134090: let C equal Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere; then a(n) = column 0 of row n of (I + D*C)^n (see A134090). - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 07 2007

a(n) = [x^n] Sum_{k=0..n} C(n,k) * x^k / [Product_{i=0..k} (1 - i*x)]; equivalently, a(n) = Sum_{k=0..n} C(n,k) * S2(n,k), where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind. - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 07 2007

A098546(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...

A036040(n) begins 1 1 1 1 3 1 1 4 3 6 1 ...

so

A122454(n) begins 1 2 1 3 9 1 4 24 18 24 1 ...

and

the present sequence begins 1 3 13 71 ...

with A000041 entries per row.

sortAbrSteg := proc(L1, L2) local i ; if nops(L1) < nops(L2) then RETURN(true) ; elif nops(L2) < nops(L1) then RETURN(false) ; else for i from 1 to nops(L1) do if op(i, L1) < op(i, L2) then RETURN(false) ; fi ; od ; RETURN(true) ; fi ; end: A098546 := proc(n, k) local prts, m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then m := nops(op(k, prts)) ; binomial(n, m) ; else 0 ; fi ; end: M3 := proc(L) local n, k, an, resul; n := add(i, i=L) ; resul := factorial(n) ; for k from 1 to n do an := add(1-min(abs(j-k), 1), j=L) ; resul := resul/ (factorial(k))^an /factorial(an) ; od ; end: A036040 := proc(n, k) local prts, m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then M3(op(k, prts)) ; else 0 ; fi ; end: A122454 := proc(n, k) A098546(n, k)*A036040(n, k) ; end: A122455 := proc(n) add(A122454(n, k), k=1..combinat[numbpart](n)) ; end: seq(A122455(n), n=1..18) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 17 2007

(PARI) a(n)= polcoeff(sum(k=0, n, binomial(n, k)*x^k/prod(i=0, k, 1-i*x +x*O(x^n))), n) - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 07 2007

Cf. A000041 A000110 A036040 A098545 A098546 A122454.

Cf. A134090; A048993 (S2).

Sequence in context: A024337 A001495 A162326 this_sequence A126390 A003319 A158882

Adjacent sequences: A122452 A122453 A122454 this_sequence A122456 A122457 A122458

easy,nonn

Alford Arnold (Alford1940(AT)aol.com), Sep 18 2006

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 17 2007