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 A088956 Triangle, read by rows, of coefficients of the hyperbinomial transform. 29
 1, 1, 1, 3, 2, 1, 16, 9, 3, 1, 125, 64, 18, 4, 1, 1296, 625, 160, 30, 5, 1, 16807, 7776, 1875, 320, 45, 6, 1, 262144, 117649, 27216, 4375, 560, 63, 7, 1, 4782969, 2097152, 470596, 72576, 8750, 896, 84, 8, 1, 100000000, 43046721, 9437184, 1411788, 163296, 15750 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The hyperbinomial transform of a sequence {b} is defined to be the sequence {d} given by d(n) = Sum_{k=0..n} T(n,k)*b(k)), where T(n,k) = (n-k+1)^(n-k-1)*C(n,k). Given a table in which the n-th row is the n-th binomial transform of the first row, then the hyperbinomial transform of any diagonal results in the next lower diagonal in the table. The simplest example of a table of iterated binomial transforms is A009998, with a main diagonal of {1,2,9,64,625,...}; and the hyperbinomial transform of this diagonal gives the next lower diagonal, {1,3,16,125,1296,...}, since 1=(1)*1, 3=(1)*1+(1)*2, 16=(3)*1+(2)*2+(1)*9, 125=(16)*1+(9)*2+(3)*9+(1)*64, etc. Another example: the hyperbinomial transform maps A065440 into A055541, since HYPERBINOMIAL([1,1,1,8,81,1024,15625]) = [1,2,6,36,320,3750,54432] where e.g.f.: A065440(x)+x = x-x/( LambertW(-x)*(1+LambertW(-x)) ), e.g.f.: A055541(x) = x-x*LambertW(-x). The m-th iteration of the hyperbinomial transform is given by the triangle of coefficients defined by T_m(n,k) = m*(n-k+m)^(n-k-1)*binomial(n,k). Example: PARI code for T_m: {a=[1,1,1,8,81,1024,15625]; m=1; b=vector(length(a)); for(n=0,length(a)-1, b[n+1]=sum(k=0,n, m*(n-k+m)^(n-k-1)*binomial(n,k)*a[k+1]); print1(b[n+1],","))} RETURNS b=[1,2,6,36,320,3750,54432]. The INVERSE hyperbinomial transform is thus given by m=-1: {a=[1,2,6,36,320,3750,54432]; m=-1; b=vector(length(a)); for(n=0,length(a)-1, b[n+1]=sum(k=0,n, m*(n-k+m)^(n-k-1)*binomial(n,k)*a[k+1]); print1(b[n+1],","))} RETURNS b=[1,1,1,8,81,1024,15625]. Simply stated, the HYPERBINOMIAL transform is to -LambertW(-x)/x as the BINOMIAL transform is to exp(x). Let A[n] be the set of all forests of labeled rooted trees on n nodes. Build a superset B[n] of A[n] by designating "some" (possibly all or none) of the isolated nodes in each forest. T(n,k) is the number of elements in B[n] with exactly k designated nodes. See A219034. - Geoffrey Critzer, Nov 10 2012 T(n,k) = A095890(n+1,k+1) * a007318(n,k) / (n-k+1), 0 <= k <= n. - Reinhard Zumkeller, Jul 07 2013 LINKS T. D. Noe, Rows n=0..50 of triangle, flattened G. Helms, Pascalmatrix tetrated E. W. Weisstein, Abel Polynomial, From MathWorld - A Wolfram Web Resource. FORMULA T(n, k) = (n-k+1)^(n-k-1)*C(n, k). E.g.f.: -LambertW(-x)*exp(x*y)/x. - Vladeta Jovovic, Oct 27 2003 From Peter Bala, Sep 11 2012: (Start) Let T(x) = Sum_{n >= 0} n^(n-1)*x^n/n! denote the tree function of A000169. The e.g.f. is (T(x)/x)*exp(t*x) = exp(T(x))*exp(t*x) = 1 + (1 + t)*x + (3 + 2*t + t^2)*x^2/2! + .... Hence the triangle is the exponential Riordan array [T(x)/x,x] belonging to the exponential Appell group. The matrix power (A088956)^r has the e.g.f. exp(r*T(x))*exp(t*x) with triangle entries given by r*(n-k+r)^(n-k-1)*binomial(n,k) for n and k >= 0. See A215534 for the case r = -1. Let A(n,x) = x*(x+n)^(n-1) be an Abel polynomial. The present triangle is the triangle of connection constants expressing A(n,x+1) as a linear combination of the basis polynomials A(k,x), 0 <= k <= n. For example, A(4,x+1) = 125*A(0,x) + 64*A(1,x) + 18*A(2,x) + 4*A(3,x) + A(4,x) gives row 4 as [125,64,18,4,1]. Let S be the array with the sequence [1,2,3,...] on the main subdiagonal and zeros elsewhere. S is the infinitesimal generator for Pascal's triangle (see A132440). Then the infinitesimal generator for this triangle is S*A088956; that is, A088956 = Exp(S*A088956), where Exp is the matrix exponential. With T(x) the tree function as above, define E(x) = T(x)/x. Then A088956 = E(S) = sum {n>=0} (n+1)^(n-1)*S^n/n!. For commuting lower unit triangular matrices A and B, we define A raised to the matrix power B, denoted A^^B, to be the matrix Exp(B*log(A)), where the matrix logarithm Log(A) is defined as sum {n >= 1} (-1)^(n+1)*(A-1)^n/n. Let P denote Pascal's triangle A007318. Then the present triangle, call it X, solves the matrix equation P^^X = X . See A215652 for the solution to X^^P = P. Furthermore, if we denote the inverse of X by Y then X^^Y = P. As an infinite tower of matrix powers, A088956 = P^^(P^^(P^^(...). A089956 augmented with the sequence (x,x,x,...) on the first superdiagonal is the production matrix for the row polynomials of A105599. (End) Sum_{k = 0..n} T(n,n-k)*(x - k - 1)^(n-k) = x^n. Setting x = n + 1 gives Sum_{k = 0..n} T(n,k)*k^k = (n + 1)^n. - Peter Bala, Feb 17 2017 EXAMPLE Rows begin:        {1},        {1,      1},        {3,      2,     1},       {16,      9,     3,    1},      {125,     64,    18,    4,   1},     {1296,    625,   160,   30,   5,  1},    {16807,   7776,  1875,  320,  45,  6, 1},   {262144, 117649, 27216, 4375, 560, 63, 7, 1}, ... MATHEMATICA nn=8; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]! CoefficientList[Series[Exp[t+y x] , {x, 0, nn}], {x, y}] //Grid (* Geoffrey Critzer, Nov 10 2012 *) PROG (Haskell) a088956 n k =  a095890 (n + 1) (k + 1) * a007318' n k `div` (n - k + 1) a088956_row n = map (a088956 n) [0..n] a088956_tabl = map a088956_row [0..] -- Reinhard Zumkeller, Jul 07 2013 CROSSREFS Cf. A088957 (row sums), A000272 (first column), A009998, A105599, A132440, A215534 (matrix inverse), A215652. Cf. A227325 (central terms). Sequence in context: A166884 A136220 A248035 * A106208 A129377 A136733 Adjacent sequences:  A088953 A088954 A088955 * A088957 A088958 A088959 KEYWORD nonn,tabl,nice AUTHOR Paul D. Hanna, Oct 26 2003 STATUS approved

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Last modified February 17 23:35 EST 2020. Contains 332006 sequences. (Running on oeis4.)