A214670 Triangle, read by rows of n*(n+1)/2 terms, where row n equals the coefficients in the series reversion of the function G(x,n)-1 such that: x = Sum_{m>=1} 1/G(x,n)^(n*m) * Product_{k=1..m} (1 - 1/G(x,n)^k), for n>=1.
1, 1, -1, -1, 1, -2, -1, 4, 4, 1, 1, -3, 0, 11, 1, -30, -42, -26, -8, -1, 1, -4, 2, 20, -19, -100, 3, 403, 808, 861, 584, 262, 76, 13, 1, 1, -5, 5, 30, -65, -191, 378, 1557, 103, -8551, -23911, -37958, -41831, -34156, -21179, -10015, -3571, -933, -169, -19, -1
Offset: 1
Examples
Consider the family of power series G(x,n) that satisfy:
x = Sum_{m>=1} 1/G(x,n)^(n*m) * Product_{k=1..m} (1 - 1/G(x,n)^k).
Examples of sequences with g.f. G(x,n) are:
n=2: A001002 = [1, 1, 1, 3, 10, 38, 154, 654, 2871, 12925, ...];
n=3: A181997 = [1, 1, 2, 9, 46, 259, 1539, 9484, 59961, ...];
n=4: A181998 = [1, 1, 3, 18, 124, 935, 7443, 61510, 522467, ...];
n=5: A209441 = [1, 1, 4, 30, 260, 2463, 24656, 256493, 2745149, ...];
n=6: A209442 = [1, 1, 5, 45, 470, 5365, 64766, 813012, 10505163, ...]; ...
Observe that Series_Reversion( G(x,n) - 1 ) is given by the polynomials:
n=1: x;
n=2: x - x^2 - x^3;
n=3: x - 2*x^2 - x^3 + 4*x^4 + 4*x^5 + x^6;
n=4: x - 3*x^2 + 11*x^4 + x^5 - 30*x^6 - 42*x^7 - 26*x^8 - 8*x^9 - x^10;
n=5: x - 4*x^2 + 2*x^3 + 20*x^4 - 19*x^5 - 100*x^6 + 3*x^7 + 403*x^8 + 808*x^9 + 861*x^10 + 584*x^11 + 262*x^12 + 76*x^13 + 13*x^14 + x^15; ...
This triangle of coefficients in the above polynomials begins:
[1];
[1, -1, -1];
[1, -2, -1, 4, 4, 1];
[1, -3, 0, 11, 1, -30, -42, -26, -8, -1];
[1, -4, 2, 20, -19, -100, 3, 403, 808, 861, 584, 262, 76, 13, 1];
[1, -5, 5, 30, -65, -191, 378, 1557, 103, -8551, -23911, -37958, -41831, -34156, -21179, -10015, -3571, -933, -169, -19, -1];
[1, -6, 9, 40, -145, -261, 1384, 2897, -8980, -38710, -14146, 258401, 990407, 2170834, 3426095, 4198850, 4137440, 3336534, 2220430, 1221799, 554027, 205250, 61206, 14351, 2550, 323, 26, 1];
[1, -7, 14, 49, -266, -245, 3325, 2596, -36710, -70556, 281645, 1413916, 1184890, -10255248, -54012830, -156371880, -329973512, -552895722, -765517470, -895408431, -896614676, -774834055, -580511469, -377792286, -213512611, -104550572, -44163315, -15985147, -4910774, -1263620, -267378, -45321, -5918, -559, -34, -1]; ...
Links
- Paul D. Hanna, Rows 1..12, flattened.
Crossrefs
Programs
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PARI
{T(n,k)=local(Axy=x*y);Axy=sum(m=1,n,-x^m*prod(j=1,m,(1-(1+y)^j)/(1-x*(1+y)^j)+x*O(x^n)));polcoeff(polcoeff(Axy,n,x),k,y)} {for(n=1,10,for(k=1,n*(n+1)/2,print1(T(n,k),", "));print(""))} -
PARI
{a(n,p)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(p*m)*prod(k=1, m, 1-1/Ser(A)^k)), #A-1)); A[n+1]} {for(n=1,8,Tn=Vec(serreverse(sum(m=1,n*(n+1)/2,a(m,n)*x^m)+x*O(x^(n*(n+1)/2))));for(k=1,n*(n+1)/2,print1(Tn[k],", "));print(""))}
Formula
G.f.: A(x,y) = Sum_{n>=1} -x^n * Product_{k=1..n} (1 - (1+y)^k) / (1 - x*(1+y)^k).
G.f. for row n is R(y,n) = Sum_{k=1..n*(n+1)/2} y^k*T(n,k) defined by:
A(x,y) = Sum_{n>=1} x^n * R(y,n) such that:
R(y,n) = Series_Reversion( G(y,n) - 1 ) where G(y,n) satisfies:
y = Sum_{m>=1} 1/G(y,n)^(n*m) * Product_{k=1..m} (1 - 1/G(y,n)^k), for n>=1.
Row polynomials R(y,n) satisfy:
(1) R(1,n) = (-1)^(n-1) * A005014(n) for n>=1.
(2) R(-1,n) = 1 for n>=1.
(3) R'(-1,n) = 0 for n>1.
(4) R'(1,n) = A214669(n) for n>=1.
Comments