A121433 Number of subpartitions of partition P=[0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,...], where P(n) = [(sqrt(8*n+49) - 7)/2].

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 13, 21, 30, 40, 51, 63, 139, 229, 334, 455, 593, 749, 924, 2043, 3378, 4951, 6785, 8904, 11333, 14098, 17226, 37971, 62655, 91728, 125671, 164997, 210252, 262016, 320904, 387567, 850260, 1397268, 2038545, 2784850, 3647788

Offset

0,6

Comments

See A115728 for the definition of subpartitions of a partition.

Links

Table of n, a(n) for n=0..44.

Formula

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^P(n), where P(n)=[(sqrt(8*n+49)-7)/2].

Example

The g.f. may be illustrated by:
1/(1-x) = (1 + x + x^2 + x^3)*(1-x)^0 +
(x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8)*(1-x)^1 +
(6*x^9 + 13*x^10 + 21*x^11 + 30*x^12 + 40*x^13 + 51*x^14)*(1-x)^2 +
(63*x^15 + 139*x^16 + 229*x^17 + 334*x^18 + 455*x^19 + 593*x^20 + 749*x^21)*(1-x)^3 +
When the sequence is put in the form of a triangle:
1, 1, 1, 1,
1, 2, 3, 4, 5,
6, 13, 21, 30, 40, 51,
63, 139, 229, 334, 455, 593, 749,
924, 2043, 3378, 4951, 6785, 8904, 11333, 14098,
17226, 37971, 62655, 91728, 125671, 164997, 210252, 262016, 320904,
then the columns of this triangle form column 3 (with offset)
of successive matrix powers of triangle H=A121412.
Column 3 of successive powers of matrix H begin:
H^1: [1,1,6,63,924,17226,387567,10182744,305379129,...];
H^2: [1,2,13,139,2043,37971,850260,22224723,663173878,...];
H^3: [1,3,21,229,3378,62655,1397268,36351147,1079567193,...];
H^4: [1,4,30,334,4951,91728,2038545,52807195,1561301733,...];
H^5: 1, [5,40,455,6785,125671,2784850,71859275,2115718545,...];
H^6: 1,6, [51,593,8904,164997,3647788,93796335,2750797677,...];
H^7: 1,7,63, [749,11333,210252,4639852,118931226,3475200792,...];
H^8: 1,8,76,924, [14098,262016,5774466,147602118,4298315847,...];
H^9: 1,9,90,1119,17226, [320904,7066029,180173970,5230303902,...];
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+49)+1)\2 - 3 ) )); polcoeff(A, n))}

Crossrefs

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121431, A121432.

Sequence in context: A181303 A276529 A200330 * A317778 A259541 A274838

Adjacent sequences: A121430 A121431 A121432 * A121434 A121435 A121436

Keyword

nonn

Author

Paul D. Hanna, Jul 30 2006

Status

approved