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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].

%I A121432 #3 Mar 30 2012 18:36:58
%S A121432 1,1,1,1,2,3,4,5,11,18,26,35,45,101,169,250,345,455,581,1305,2190,
%T A121432 3255,4520,6006,7735,9730,21745,36360,53916,74781,99351,128051,161336,
%U A121432 199692,443329,737051,1087583,1502270,1989113,2556806,3214774,3973212,4843125
%N 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].
%C A121432 See A115728 for the definition of subpartitions of a partition.
%F A121432 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].
%e A121432 The g.f. may be illustrated by:
%e A121432 1/(1-x) = (1 + x + x^2)*(1-x)^0 + (x^3 + 2*x^4 + 3*x^5 + 4*x^6)*(1-x)^1 +
%e A121432 (5*x^7 + 11*x^8 + 18*x^9 + 26*x^10 + 35*x^11)*(1-x)^2 +
%e A121432 (45*x^12 + 101*x^13 + 169*x^14 + 250*x^15 + 345*x^16 + 455*x^17)*(1-x)^3 +
%e A121432 (581*x^18 + 1305*x^19 + 2190*x^20 + 3255*x^21 + 4520*x^22 + 6006*x^23 + 7735*x^24)*(1-x)^4 +...
%e A121432 When the sequence is put in the form of a triangle:
%e A121432 1, 1, 1,
%e A121432 1, 2, 3, 4,
%e A121432 5, 11, 18, 26, 35,
%e A121432 45, 101, 169, 250, 345, 455,
%e A121432 581, 1305, 2190, 3255, 4520, 6006, 7735,
%e A121432 9730, 21745, 36360, 53916, 74781, 99351, 128051, 161336, ...
%e A121432 then the columns of this triangle form column 2 (with offset)
%e A121432 of successive matrix powers of triangle H=A121412.
%e A121432 This sequence is embedded in table A121428 as follows.
%e A121432 Column 2 of successive powers of matrix H begin:
%e A121432 H^1: [1,1,5,45,581,9730,199692,4843125,135345925,...];
%e A121432 H^2: [1,2,11,101,1305,21745,443329,10679494,296547736,...];
%e A121432 H^3: [1,3,18,169,2190,36360,737051,17645187,487025244,...];
%e A121432 H^4: 1, [4,26,250,3255,53916,1087583,25889969,710546530,...];
%e A121432 H^5: 1,5, [35,345,4520,74781,1502270,35578270,971255050,...];
%e A121432 H^6: 1,6,45, [455,6006,99351,1989113,46890210,1273698270,...];
%e A121432 H^7: 1,7,56,581, [7735,128051,2556806,60022670,1622857887,...];
%e A121432 H^8: 1,8,68,724,9730, [161336,3214774,75190410,2024181693,...];
%e A121432 H^9: 1,9,81,885,12015,199692, [3973212,92627235,2483617140,...];
%e A121432 the terms enclosed in brackets form this sequence.
%o A121432 (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))}
%Y A121432 Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121428, A121429; column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121431, A121433.
%K A121432 nonn
%O A121432 0,5
%A A121432 _Paul D. Hanna_, Jul 30 2006