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A088956 Triangle, read by rows, of coefficients of the hyperbinomial transform.

%I A088956 #53 Apr 05 2025 19:53:12
%S A088956 1,1,1,3,2,1,16,9,3,1,125,64,18,4,1,1296,625,160,30,5,1,16807,7776,
%T A088956 1875,320,45,6,1,262144,117649,27216,4375,560,63,7,1,4782969,2097152,
%U A088956 470596,72576,8750,896,84,8,1,100000000,43046721,9437184,1411788,163296,15750
%N A088956 Triangle, read by rows, of coefficients of the hyperbinomial transform.
%C A088956 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).
%C A088956 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.
%C A088956 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.
%C A088956 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).
%C A088956 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).
%C A088956 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].
%C A088956 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].
%C A088956 Simply stated, the HYPERBINOMIAL transform is to -LambertW(-x)/x as the BINOMIAL transform is to exp(x).
%C A088956 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
%H A088956 T. D. Noe, <a href="/A088956/b088956.txt">Rows n=0..50 of triangle, flattened</a>
%H A088956 T. Copeland, <a href="https://tcjpn.wordpress.com/2021/11/20/composition-conjugation-and-the-umbral-calculus-part-i/">Composition, Conjugation, and the Umbral Calculus-Part I</a>, 2021.
%H A088956 G. Helms, <a href="http://go.helms-net.de/math/tetdocs/PascalMatrixTetrated.pdf">Pascalmatrix tetrated</a>
%H A088956 E. W. Weisstein, <a href="https://mathworld.wolfram.com/AbelPolynomial.html">Abel Polynomial</a>, From MathWorld - A Wolfram Web Resource.
%H A088956 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A088956 T(n, k) = (n-k+1)^(n-k-1)*C(n, k).
%F A088956 E.g.f.: -LambertW(-x)*exp(x*y)/x. - _Vladeta Jovovic_, Oct 27 2003
%F A088956 From _Peter Bala_, Sep 11 2012: (Start)
%F A088956 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.
%F A088956 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.
%F A088956 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].
%F A088956 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.
%F A088956 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!.
%F A088956 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^^(...))).
%F A088956 A088956 augmented with the sequence (x,x,x,...) on the first superdiagonal is the production matrix for the row polynomials of A105599.
%F A088956 (End)
%F A088956 T(n,k) = A095890(n+1,k+1) * A007318(n,k) / (n-k+1), 0 <= k <= n. - _Reinhard Zumkeller_, Jul 07 2013
%F A088956 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
%F A088956 As lower triangular matrices, this entry, T, equals unsigned A137542 * A007318 * signed A059297. The Pascal matrix is sandwiched between a pair of inverse matrices, so this entry is conjugate to the Pascal matrix, allowing convergent analytic expressions of T, say f(T), to be computed as f(A007318) sandwiched between the inverse pair. - _Tom Copeland_, Dec 06 2021
%e A088956 Rows begin:
%e A088956        {1},
%e A088956        {1,      1},
%e A088956        {3,      2,     1},
%e A088956       {16,      9,     3,    1},
%e A088956      {125,     64,    18,    4,   1},
%e A088956     {1296,    625,   160,   30,   5,  1},
%e A088956    {16807,   7776,  1875,  320,  45,  6, 1},
%e A088956   {262144, 117649, 27216, 4375, 560, 63, 7, 1}, ...
%t A088956 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 *)
%o A088956 (Haskell)
%o A088956 a088956 n k =  a095890 (n + 1) (k + 1) * a007318' n k `div` (n - k + 1)
%o A088956 a088956_row n = map (a088956 n) [0..n]
%o A088956 a088956_tabl = map a088956_row [0..]
%o A088956 -- _Reinhard Zumkeller_, Jul 07 2013
%Y A088956 Cf. A088957 (row sums), A000272 (first column), A009998, A105599, A132440, A215534 (matrix inverse), A215652.
%Y A088956 Cf. A227325 (central terms).
%Y A088956 Cf. A007318, A059297, A137542.
%K A088956 nonn,tabl,nice
%O A088956 0,4
%A A088956 _Paul D. Hanna_, Oct 26 2003