cp's OEIS Frontend

A086606 Triangle, read by rows, where the n-th row is the first n terms of the n-th self-convolution of the sequence formed by flattening this triangle.

%I A086606 #9 Apr 18 2012 06:20:26
%S A086606 1,1,2,1,3,9,1,4,14,32,1,5,20,55,140,1,6,27,86,243,630,1,7,35,126,392,
%T A086606 1099,2870,1,8,44,176,598,1808,5048,13256,1,9,54,237,873,2835,8433,
%U A086606 23454,61389,1,10,65,310,1230,4272,13495,39640,109400,286710,1,11,77
%N A086606 Triangle, read by rows, where the n-th row is the first n terms of the n-th self-convolution of the sequence formed by flattening this triangle.
%H A086606 Paul D. Hanna, <a href="/A086606/b086606.txt">Table of n, a(n) for n = 0..1035</a>
%e A086606 This triangle begins:
%e A086606 1;
%e A086606 1, 2;
%e A086606 1, 3,  9;
%e A086606 1, 4, 14,  32;
%e A086606 1, 5, 20,  55, 140;
%e A086606 1, 6, 27,  86, 243,  630;
%e A086606 1, 7, 35, 126, 392, 1099, 2870;
%e A086606 1, 8, 44, 176, 598, 1808, 5048, 13256; ...
%e A086606 The g.f. A(x) of this sequence as a flat list of coefficients begins:
%e A086606 A(x) = 1 + x + 2*x^2 + x^3 + 3*x^4 + 9*x^5 + x^6 + 4*x^7 + 14*x^8 + 32*x^9 + x^10 + 5*x^11 + 20*x^12 + 55*x^13 + 140*x^14 +...
%e A086606 such that the coefficients in A(x)^n, n>=1, forms the table:
%e A086606 A^1: [(1),1,  2,   1,   3,    9,    1,     4,    14,     32, ...];
%e A086606 A^2: [(1, 2), 5,   6,  12,   28,   33,    52,    67,    164, ...];
%e A086606 A^3: [(1, 3,  9), 16,  33,   72,  125,   222,   330,    646, ...];
%e A086606 A^4: [(1, 4, 14,  32), 73,  164,  334,   660,  1152,   2184, ...];
%e A086606 A^5: [(1, 5, 20,  55, 140), 336,  755,  1625,  3195,   6315, ...];
%e A086606 A^6: [(1, 6, 27,  86, 243,  630),1532,  3546,  7635,  16020, ...];
%e A086606 A^7: [(1, 7, 35, 126, 392, 1099, 2870), 7092, 16443,  36666, ...];
%e A086606 A^8: [(1, 8, 44, 176, 598, 1808, 5048, 13256),32761,  77384, ...];
%e A086606 A^9: [(1, 9, 54, 237, 873, 2835, 8433, 23454, 61389),153007, ...]; ...
%e A086606 where the lower triangular portion equals this sequence.
%o A086606 (PARI) /* As a flattened triangle: */
%o A086606 {a(n)=local(t=(sqrt(8*n+1)+1)\2,A=1+sum(k=1,min(n-1,t),a(k)*x^k));if(n==0,1,polcoeff((A+x*O(x^n))^t,n-t*(t-1)/2))}
%o A086606 for(n=0,60,print1(a(n),", "))
%Y A086606 Cf. A086607 (main diagonal), A086608 (row sums).
%K A086606 nonn,tabl
%O A086606 0,3
%A A086606 _Paul D. Hanna_, Jul 23 2003