This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS The row sums are a signed version of A005014. [From _Olivier Gérard_, Jun 26 2012, in an email to the seqfan list, which suggested that the g.f. A(x,y) is a generalization of the g.f. for A005014.] LINKS Paul D. Hanna, Rows 1..12, flattened. 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. EXAMPLE 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]; ... PROG (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(""))} CROSSREFS Cf. A001002, A181997, A181998, A209441, A209442, A005014 (row sums), A214669. Cf. A214690 (variant). Sequence in context: A105568 A004175 A136756 * A181878 A256791 A274980 Adjacent sequences:  A214667 A214668 A214669 * A214671 A214672 A214673 KEYWORD sign,tabf AUTHOR Paul D. Hanna, Jul 25 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.