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A121432 Number of subpartitions of partition P=[0,0,0,1,1,1,1,2,2,2,2,2,3,...], where P(n) = [(sqrt(8*n+25) - 5)/2]. 4
1, 1, 1, 1, 2, 3, 4, 5, 11, 18, 26, 35, 45, 101, 169, 250, 345, 455, 581, 1305, 2190, 3255, 4520, 6006, 7735, 9730, 21745, 36360, 53916, 74781, 99351, 128051, 161336, 199692, 443329, 737051, 1087583, 1502270, 1989113, 2556806, 3214774, 3973212, 4843125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

See A115728 for the definition of subpartitions of a partition.

LINKS

Table of n, a(n) for n=0..42.

FORMULA

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^P(n), where P(n)=[(sqrt(8*n+25)-5)/2].

EXAMPLE

The g.f. may be illustrated by:

1/(1-x) = (1 + x + x^2)*(1-x)^0 + (x^3 + 2*x^4 + 3*x^5 + 4*x^6)*(1-x)^1 +

(5*x^7 + 11*x^8 + 18*x^9 + 26*x^10 + 35*x^11)*(1-x)^2 +

(45*x^12 + 101*x^13 + 169*x^14 + 250*x^15 + 345*x^16 + 455*x^17)*(1-x)^3 +

(581*x^18 + 1305*x^19 + 2190*x^20 + 3255*x^21 + 4520*x^22 + 6006*x^23 + 7735*x^24)*(1-x)^4 +...

When the sequence is put in the form of a triangle:

1, 1, 1,

1, 2, 3, 4,

5, 11, 18, 26, 35,

45, 101, 169, 250, 345, 455,

581, 1305, 2190, 3255, 4520, 6006, 7735,

9730, 21745, 36360, 53916, 74781, 99351, 128051, 161336, ...

then the columns of this triangle form column 2 (with offset)

of successive matrix powers of triangle H=A121412.

This sequence is embedded in table A121428 as follows.

Column 2 of successive powers of matrix H begin:

H^1: [1,1,5,45,581,9730,199692,4843125,135345925,...];

H^2: [1,2,11,101,1305,21745,443329,10679494,296547736,...];

H^3: [1,3,18,169,2190,36360,737051,17645187,487025244,...];

H^4: 1, [4,26,250,3255,53916,1087583,25889969,710546530,...];

H^5: 1,5, [35,345,4520,74781,1502270,35578270,971255050,...];

H^6: 1,6,45, [455,6006,99351,1989113,46890210,1273698270,...];

H^7: 1,7,56,581, [7735,128051,2556806,60022670,1622857887,...];

H^8: 1,8,68,724,9730, [161336,3214774,75190410,2024181693,...];

H^9: 1,9,81,885,12015,199692, [3973212,92627235,2483617140,...];

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+25)+1)\2 - 2 ) )); polcoeff(A, n))}

CROSSREFS

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121428, A121429; column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121431, A121433.

Sequence in context: A333610 A052418 A051800 * A118968 A073528 A116067

Adjacent sequences:  A121429 A121430 A121431 * A121433 A121434 A121435

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 30 2006

STATUS

approved

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Last modified September 20 02:19 EDT 2021. Contains 347577 sequences. (Running on oeis4.)