1,5

Sum of the n-th row is the number of all permutations of n elements: Sum_{k=1..n, T(n,k)} = n! = A000142(n) We can extend T(n,k)=0, if k<=0 or k>n.

Contribution from Peter Luschny (peter(AT)luschny.de), Mar 07 2009: (Start)

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -1,

summed over parts with equal biggest part (see the Luschny link).

Underlying partition triangle is A102189.

Same partition product with length statistic is A008275.

Diagonal a(A000217(n)) = rising_factorial(1,n-1), A000142(n-1) (n > 0).

Row sum is A000142. (End)

Let k in {1,2,3,...} index the family of sequences A000012,A000085,A057693, A070945,A070946,A070947,... respectively. Column k is the kth sequence minus its immediate predecessor. For example, T(5,3)=A057693(5)-A000085(5) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 23 2009]

IBM Research : Ponder This

Peter Luschny, Counting with Partitions. [From Peter Luschny (peter(AT)luschny.de), Mar 07 2009]

Peter Luschny, Generalized Stirling_1 Triangles. [From Peter Luschny (peter(AT)luschny.de), Mar 07 2009]

T(n,1) = 1 T(n,2) = n! * Sum_{k=1..[n/2], (1/(k! * (2!)^k * (n-2k)!)} T(n,k) = n!/k * (1-1/(n-k)-...-1/(k+1)-1/2k), if n/3 < k <= n/2 T(n,k) = n!/k, if n/2 < k <= n T(n,n) = (n-1)! = A000142(n-1)

E.g.f. for k-th column: exp(-x^k*LerchPhi(x,1,k))*(exp(x^k/k)-1)/(1-x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 03 2007

Contribution from Peter Luschny (peter(AT)luschny.de), Mar 07 2009: (Start)

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n

T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that

1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),

f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n+1). (End)

Table[CoefficientList[ Series[(Exp[x^m/m] - 1) Exp[Sum[x^k/k, {k, 1, m - 1}]], {x, 0, 8}], x]*Table[n!, {n, 0, 8}], {m, 1, 8}] // Transpose // Grid [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 23 2009]

Cf. A000142.

Cf. A071007, A080510.

Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395 [From Peter Luschny (peter(AT)luschny.de), Mar 07 2009]

Sequence in context: A155788 A108073 A057731 this_sequence A108916 A119421 A121581

Adjacent sequences: A126071 A126072 A126073 this_sequence A126075 A126076 A126077

base,nonn,tabl

Dan Dima (dimad72(AT)gmail.com), Mar 01 2007