|
|
EXAMPLE
|
G.f.: x = 1*x/(1+x) + 1*x^2/(1+x)^8 + 7*x^3/(1+x)^27 + 154*x^4/(1+x)^64 + 7329*x^5/(1+x)^125 + 621054*x^6/(1+x)^216 + 83287785*x^7/(1+x)^343 +...
ALTERNATE GENERATING METHOD.
Also forms the final terms in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated 3*n*(n-1)+1 times, starting with a '1' in row 1, as illustrated by:
1;
1, 1, 1, 1, 1, 1, 1;
1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7;
1, 3, 6, 10, 15, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154; ...
MATRIX GENERATING METHOD.
Given triangle T(n, k) = binomial(k^3+n-k-1, n-k), such that the g.f. of column k equals 1/(1-x)^(k^3) for k>=1, which begins:
1;
1, 1;
1, 8, 1;
1, 36, 27, 1;
1, 120, 378, 64, 1;
1, 330, 3654, 2080, 125, 1;
1, 792, 27405, 45760, 7875, 216, 1;
1, 1716, 169911, 766480, 333375, 23436, 343, 1; ...
then this sequence forms column 1 (ignoring signs) of the matrix inverse:
1;
-1, 1;
7, -8, 1;
-154, 180, -27, 1;
7329, -8616, 1350, -64, 1;
-621054, 731502, -116244, 5920, -125, 1;
83287785, -98171784, 15685569, -820480, 19125, -216, 1;
-16339143828, 19265191212, -3085386984, 163253040, -3963750, 50652, -343, 1; ...
|
|
|
PROG
|
(PARI) /* GENERATING FUNCTION: */
{a(n)=local(F=1/(1+x+x*O(x^n))); polcoeff(x-sum(k=1, n-1, a(k)*x^k*F^(k^3)), n)}
for(n=1, 20, print1(a(n), ", "))
(PARI) /* SUMMATION METHOD: */
{A=[1, 1]; for(i=1, 20, A=concat(A, -Vec(sum(n=0, #A-1, A[n+1]*x^n/(1+x+x*O(x^#A))^(n^3)))[#A+1])); for(n=1, #A-1, print1(A[n+1], ", "))}
(PARI) /* MATRIX METHOD: */
{a(n)=local(M=matrix(n, n, r, c, if(r>=c, binomial(c^3+r-c-1, r-c)))); -(-1)^n*(M^-1)[n, 1]}
for(n=1, 20, print1(a(n), ", "))
|
