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A091492 Triangle, read by rows, generated recursively and related to partitions. 3
1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 1, 1, 3, 1, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 0, 0, 0, 1, 1, 4, 3, 0, 0, 0, 0, 0, 0, 1, 1, 4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 1, 5, 5, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 5, 7, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 8, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,18

COMMENTS

Excluding the leading zeros, the columns are related to partitions. The 3rd column lists A001399 (partitions of n into at most 3 parts). The 4th column lists A001400 (partitions of n into at most 4 parts). The 5th column lists A001401 (partitions of n into at most 5 parts). The 6th column is A091498. Row sums are A091493. The number of nonzero terms in each row is A091497.

LINKS

Table of n, a(n) for n=0..104.

FORMULA

T(n, k)=Sum T(n-k, j)*T(j, k-j) {j=[(k+1)/2]..min(k, n-k)}, with T(0, 0)=1, T(n, 0)=1, T(1, 1)=1.

EXAMPLE

T(12,3) = 7 = (4)*1+(3)*1 = T(9,2)*T(2,1)+T(9,3)*T(3,0) = Sum T(9,j)*T(j,3-j) {j=2..3}.

Rows begin:

{1},

{1,1},

{1,1,0},

{1,1,1,0},

{1,1,1,0,0},

{1,1,2,0,0,0},

{1,1,2,1,0,0,0},

{1,1,3,1,0,0,0,0},

{1,1,3,2,0,0,0,0,0},

{1,1,4,3,0,0,0,0,0,0},

{1,1,4,4,1,0,0,0,0,0,0},

{1,1,5,5,1,1,0,0,0,0,0,0},

{1,1,5,7,2,1,0,0,0,0,0,0,0},

{1,1,6,8,3,2,0,0,0,0,0,0,0,0},

{1,1,6,10,5,3,0,0,0,0,0,...

{1,1,7,12,6,5,0,0,0,0,0,...

{1,1,7,14,9,7,1,0,0,0,0,...

{1,1,8,16,11,10,2,0,0,0,...

{1,1,8,19,15,13,3,2,0,0,...

{1,1,9,21,18,18,5,2,0,0,...

{1,1,9,24,23,23,8,4,0,0,...

{1,1,10,27,27,30,11,6,0,...

{1,1,10,30,34,37,17,10,0,...

PROG

(PARI) T(n, k)=if(k>n || n<0 || k<0, 0, if(k<=1 || (k==n && n<2), 1, sum(j=(k+1)\2, min(n-k, k), T(n-k, j)*T(j, k-j)); ); )

CROSSREFS

Cf. A001399, A001400, A001401, A091493, A091497, A091498.

Sequence in context: A014081 A091890 A029431 * A284264 A273302 A025086

Adjacent sequences:  A091489 A091490 A091491 * A091493 A091494 A091495

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jan 16 2004

STATUS

approved

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Last modified June 25 07:53 EDT 2019. Contains 324347 sequences. (Running on oeis4.)