A121431 Number of subpartitions of partition P=[0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,...] (A052146).

1, 1, 1, 2, 3, 4, 9, 15, 22, 30, 69, 118, 178, 250, 335, 769, 1317, 1995, 2820, 3810, 4984, 11346, 19311, 29126, 41061, 55410, 72492, 92652, 208914, 352636, 528097, 740035, 993678, 1294776, 1649634, 2065146, 4613976, 7722840, 11476963, 15971180

Offset

0,4

Comments

See A115728 for the definition of subpartitions of a partition.

Links

Table of n, a(n) for n=0..39.

Formula

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^A052146(n).

Example

The g.f. may be illustrated by:
1/(1-x) = (1 + 1*x)*(1-x)^0 + (x^2 + 2*x^3 + 3*x^4)*(1-x)^1 +
(4*x^5 + 9*x^6 + 15*x^7 + 22*x^8)*(1-x)^2 +
(30*x^9 + 69*x^10 + 118*x^11 + 178*x^12 + 250*x^13)*(1-x)^3 +
(335*x^14 + 769*x^15 + 1317*x^16 + 1995*x^17 + 2820*x^18 + 3810*x^19)*(1-x)^4 +...
When the sequence is put in the form of a triangle:
1, 1,
1, 2, 3,
4, 9, 15, 22,
30, 69, 118, 178, 250,
335, 769, 1317, 1995, 2820, 3810,
4984, 11346, 19311, 29126, 41061, 55410, 72492,
92652, 208914, 352636, 528097, 740035, 993678, 1294776, ...
then the columns of this triangle form column 1 (with offset)
of successive matrix powers of triangle H=A121412.
This sequence is embedded in table A121426 as follows.
Column 1 of successive powers of matrix H begin:
H^1: [1,1,4,30,335,4984,92652,2065146,53636520,...];
H^2: [1,2,9,69,769,11346,208914,4613976,118840164,...];
H^3: 1, [3,15,118,1317,19311,352636,7722840,197354133,...];
H^4: 1,4, [22,178,1995,29126,528097,11476963,291124693,...];
H^5: 1,5,30, [250,2820,41061,740035,15971180,402319275,...];
H^6: 1,6,39,335, [3810,55410,993678,21310710,533345745,...];
H^7: 1,7,49,434,4984, [72492,1294776,27611970,686872893,...];
H^8: 1,8,60,548,6362,92652, [1649634,35003430,865852191,...];
H^9: 1,9,72,678,7965,116262,2065146, [43626510,1073540871,...];
the terms enclosed in brackets form this sequence.

Prog

(PARI) {a(n)=local(A); if(n==0, 1, A=x+x*O(x^n); for(k=0, n, A+=polcoeff(A, k)*x^k*(1-(1-x)^( (sqrtint(8*k+9)+1)\2 - 1 ) )); polcoeff(A, n))}

Crossrefs

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121426, A121427; column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121432, A121433.

Sequence in context: A032982 A288856 A033076 * A084080 A124753 A248647

Adjacent sequences: A121428 A121429 A121430 * A121432 A121433 A121434

Keyword

nonn

Author

Paul D. Hanna, Jul 30 2006

Status

approved