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.
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
Examples
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.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..1035
Programs
-
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),", "))