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

%I A121431 #3 Mar 30 2012 18:36:58
%S A121431 1,1,1,2,3,4,9,15,22,30,69,118,178,250,335,769,1317,1995,2820,3810,
%T A121431 4984,11346,19311,29126,41061,55410,72492,92652,208914,352636,528097,
%U A121431 740035,993678,1294776,1649634,2065146,4613976,7722840,11476963,15971180
%N A121431 Number of subpartitions of partition P=[0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,...] (A052146).
%C A121431 See A115728 for the definition of subpartitions of a partition.
%F A121431 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^A052146(n).
%e A121431 The g.f. may be illustrated by:
%e A121431 1/(1-x) = (1 + 1*x)*(1-x)^0 + (x^2 + 2*x^3 + 3*x^4)*(1-x)^1 +
%e A121431 (4*x^5 + 9*x^6 + 15*x^7 + 22*x^8)*(1-x)^2 +
%e A121431 (30*x^9 + 69*x^10 + 118*x^11 + 178*x^12 + 250*x^13)*(1-x)^3 +
%e A121431 (335*x^14 + 769*x^15 + 1317*x^16 + 1995*x^17 + 2820*x^18 + 3810*x^19)*(1-x)^4 +...
%e A121431 When the sequence is put in the form of a triangle:
%e A121431 1, 1,
%e A121431 1, 2, 3,
%e A121431 4, 9, 15, 22,
%e A121431 30, 69, 118, 178, 250,
%e A121431 335, 769, 1317, 1995, 2820, 3810,
%e A121431 4984, 11346, 19311, 29126, 41061, 55410, 72492,
%e A121431 92652, 208914, 352636, 528097, 740035, 993678, 1294776, ...
%e A121431 then the columns of this triangle form column 1 (with offset)
%e A121431 of successive matrix powers of triangle H=A121412.
%e A121431 This sequence is embedded in table A121426 as follows.
%e A121431 Column 1 of successive powers of matrix H begin:
%e A121431 H^1: [1,1,4,30,335,4984,92652,2065146,53636520,...];
%e A121431 H^2: [1,2,9,69,769,11346,208914,4613976,118840164,...];
%e A121431 H^3: 1, [3,15,118,1317,19311,352636,7722840,197354133,...];
%e A121431 H^4: 1,4, [22,178,1995,29126,528097,11476963,291124693,...];
%e A121431 H^5: 1,5,30, [250,2820,41061,740035,15971180,402319275,...];
%e A121431 H^6: 1,6,39,335, [3810,55410,993678,21310710,533345745,...];
%e A121431 H^7: 1,7,49,434,4984, [72492,1294776,27611970,686872893,...];
%e A121431 H^8: 1,8,60,548,6362,92652, [1649634,35003430,865852191,...];
%e A121431 H^9: 1,9,72,678,7965,116262,2065146, [43626510,1073540871,...];
%e A121431 the terms enclosed in brackets form this sequence.
%o A121431 (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))}
%Y A121431 Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121426, A121427; column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121432, A121433.
%K A121431 nonn
%O A121431 0,4
%A A121431 _Paul D. Hanna_, Jul 30 2006