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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.
3
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, 1099, 2870, 1, 8, 44, 176, 598, 1808, 5048, 13256, 1, 9, 54, 237, 873, 2835, 8433, 23454, 61389, 1, 10, 65, 310, 1230, 4272, 13495, 39640, 109400, 286710, 1, 11, 77
OFFSET
0,3
LINKS
EXAMPLE
This triangle begins:
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, 1099, 2870;
1, 8, 44, 176, 598, 1808, 5048, 13256; ...
The g.f. A(x) of this sequence as a flat list of coefficients begins:
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 +...
such that the coefficients in A(x)^n, n>=1, forms the table:
A^1: [(1),1, 2, 1, 3, 9, 1, 4, 14, 32, ...];
A^2: [(1, 2), 5, 6, 12, 28, 33, 52, 67, 164, ...];
A^3: [(1, 3, 9), 16, 33, 72, 125, 222, 330, 646, ...];
A^4: [(1, 4, 14, 32), 73, 164, 334, 660, 1152, 2184, ...];
A^5: [(1, 5, 20, 55, 140), 336, 755, 1625, 3195, 6315, ...];
A^6: [(1, 6, 27, 86, 243, 630),1532, 3546, 7635, 16020, ...];
A^7: [(1, 7, 35, 126, 392, 1099, 2870), 7092, 16443, 36666, ...];
A^8: [(1, 8, 44, 176, 598, 1808, 5048, 13256),32761, 77384, ...];
A^9: [(1, 9, 54, 237, 873, 2835, 8433, 23454, 61389),153007, ...]; ...
where the lower triangular portion equals this sequence.
PROG
(PARI) /* As a flattened triangle: */
{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))}
for(n=0, 60, print1(a(n), ", "))
CROSSREFS
Cf. A086607 (main diagonal), A086608 (row sums).
Sequence in context: A076655 A236438 A101486 * A076112 A208744 A122454
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jul 23 2003
STATUS
approved