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%I #3 Mar 30 2012 18:36:58
%S 1,1,1,1,1,2,3,4,5,6,13,21,30,40,51,63,139,229,334,455,593,749,924,
%T 2043,3378,4951,6785,8904,11333,14098,17226,37971,62655,91728,125671,
%U 164997,210252,262016,320904,387567,850260,1397268,2038545,2784850,3647788
%N 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].
%C See A115728 for the definition of subpartitions of a partition.
%F 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].
%e The g.f. may be illustrated by:
%e 1/(1-x) = (1 + x + x^2 + x^3)*(1-x)^0 +
%e (x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8)*(1-x)^1 +
%e (6*x^9 + 13*x^10 + 21*x^11 + 30*x^12 + 40*x^13 + 51*x^14)*(1-x)^2 +
%e (63*x^15 + 139*x^16 + 229*x^17 + 334*x^18 + 455*x^19 + 593*x^20 + 749*x^21)*(1-x)^3 +
%e When the sequence is put in the form of a triangle:
%e 1, 1, 1, 1,
%e 1, 2, 3, 4, 5,
%e 6, 13, 21, 30, 40, 51,
%e 63, 139, 229, 334, 455, 593, 749,
%e 924, 2043, 3378, 4951, 6785, 8904, 11333, 14098,
%e 17226, 37971, 62655, 91728, 125671, 164997, 210252, 262016, 320904,
%e then the columns of this triangle form column 3 (with offset)
%e of successive matrix powers of triangle H=A121412.
%e Column 3 of successive powers of matrix H begin:
%e H^1: [1,1,6,63,924,17226,387567,10182744,305379129,...];
%e H^2: [1,2,13,139,2043,37971,850260,22224723,663173878,...];
%e H^3: [1,3,21,229,3378,62655,1397268,36351147,1079567193,...];
%e H^4: [1,4,30,334,4951,91728,2038545,52807195,1561301733,...];
%e H^5: 1, [5,40,455,6785,125671,2784850,71859275,2115718545,...];
%e H^6: 1,6, [51,593,8904,164997,3647788,93796335,2750797677,...];
%e H^7: 1,7,63, [749,11333,210252,4639852,118931226,3475200792,...];
%e H^8: 1,8,76,924, [14098,262016,5774466,147602118,4298315847,...];
%e H^9: 1,9,90,1119,17226, [320904,7066029,180173970,5230303902,...];
%e the terms enclosed in brackets form this sequence.
%o (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))}
%Y Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121431, A121432.
%K nonn
%O 0,6
%A _Paul D. Hanna_, Jul 30 2006
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