A137542 -id:A137542 - cp's OEIS frontend

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

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

Offset: 0

Paul D. Hanna, Oct 26 2003

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

			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}, ...

T. D. Noe, Rows n=0..50 of triangle, flattened
T. Copeland, Composition, Conjugation, and the Umbral Calculus-Part I, 2021.
G. Helms, Pascalmatrix tetrated
E. W. Weisstein, Abel Polynomial, From MathWorld - A Wolfram Web Resource.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A088957 (row sums), A000272 (first column), A009998, A105599, A132440, A215534 (matrix inverse), A215652.
Cf. A227325 (central terms).
Cf. A007318, A059297, A137542.

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

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 *)

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^^(...))).
A088956 augmented with the sequence (x,x,x,...) on the first superdiagonal is the production matrix for the row polynomials of A105599.
(End)
T(n,k) = A095890(n+1,k+1) * A007318(n,k) / (n-k+1), 0 <= k <= n. - Reinhard Zumkeller, Jul 07 2013
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
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

A117107 Number of permutations in S_n avoiding 21{bar 3}54 (i.e., every occurrence of 2154 is contained in an occurrence of a 21354) and such that the graph corresponding to the permutation is connected (see "Forest-like permutations" below).

Original entry on oeis.org

1, 1, 3, 12, 57, 304, 1765, 10943, 71519, 488186, 3456526, 25251479, 189545179

Offset: 1

Steve Butler, Apr 18 2006

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

			See example in A137546.

M. Bousquet-Mélou and S. Butler, Forest-like permutations, arXiv:math/0603617 [math.CO], 2006.
Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.

Cf. A137536, A137537, A137538, A137539, A137540, A137541, A137542, A137543, A137544, A137545, A137546, A137547, A137548.

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

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