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A136215 Triangle T, read by rows, where T(n,k) = A007559(n-k)*C(n,k) where A007559 equals the triple factorials in column 0.

%I A136215 #22 Jun 12 2018 10:22:28
%S A136215 1,1,1,4,2,1,28,12,3,1,280,112,24,4,1,3640,1400,280,40,5,1,58240,
%T A136215 21840,4200,560,60,6,1,1106560,407680,76440,9800,980,84,7,1,24344320,
%U A136215 8852480,1630720,203840,19600,1568,112,8,1,608608000,219098880,39836160
%N A136215 Triangle T, read by rows, where T(n,k) = A007559(n-k)*C(n,k) where A007559 equals the triple factorials in column 0.
%C A136215 Comments from _Peter Bala_, Jul 10 2008: (Start) This array is the particular case P(1,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown below
%C A136215 n\k|0....................1...............2.........3.....4
%C A136215 ----------------------------------------------------------
%C A136215 0..|1.....................................................
%C A136215 1..|a....................1................................
%C A136215 2..|a(a+b)...............2a..............1................
%C A136215 3..|a(a+b)(a+2b).........3a(a+b).........3a........1......
%C A136215 4..|a(a+b)(a+2b)(a+3b)...4a(a+b)(a+2b)...6a(a+b)...4a....1
%C A136215 ...
%C A136215 See A094587 for some general properties of these arrays.
%C A136215 Other cases recorded in the database include: P(1,0) = Pascal's triangle A007318, P(1,1) = A094587, P(2,0) = A038207, P(3,0) = A027465, P(2,1) = A132159 and P(2,3) = A136216. (End)
%C A136215 The generalized Pascal matrix that Bala refers to is itself a special case of application of the formalism of A133314 to fundamental matrices derived from infinitesimal generators described in A133314, of which the fundamental Pascal (A007318), unsigned Lah (A105278) and associated Laguerre (A135278) matrices are special examples. The formalism gives, among other relations, the inverse of T as TI(n,k) = b(n-k)*C(n,k) where the sequence b is given by the list partition transform (A133314) of A007559; i.e., b = LPT(A007559) = (1,-A008544)= (1,-1,-2,-10,-80,...). The formalism of A132382 may also be applied with the double factorial A001147 replaced by the triple factorial A007559 (see also A133480). - _Tom Copeland_, Aug 18 2008
%C A136215 From _Peter Bala_, Aug 29 2013: (Start)
%C A136215 Exponential Riordan array [1/(1 - 3*y)^(1/3), y]. The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
%C A136215 Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x).
%C A136215 For example, row 3 is (28, 12, 3, 1) so P(3,x + 1) = (x + 1)*(x + 4)*(x + 7) = 28 + 12*x + 3*x*(x + 3) + x*(x + 3)*(x + 6). (End)
%H A136215 G. C. Greubel, <a href="/A136215/b136215.txt">Rows n=0..100 of triangle, flattened</a>
%H A136215 M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3
%H A136215 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sheffer_sequence">Sheffer sequence</a>
%F A136215 Column k of T = column 0 of U^(k+1) (matrix power) for k>=0 where U = A136214. Matrix square equals A136216, where A136216(n,k) = A008544(n-k)*C(n,k) where A008544 are also triple factorials.
%F A136215 From _Peter Bala_, Jul 10 2008: (Start)
%F A136215 T(n,k) = (3*n-3*k-2)*T(n-1,k) + T(n-1,k-1).
%F A136215 E.g.f. exp(x*y)/(1-3*y)^(1/3) = 1 + (1+x)*y + (4+2*x+x^2)*y^2/2! + ... . (End)
%e A136215 Column k of T = column 0 of U^(k+1), while
%e A136215 column k of U = column 0 of T^(3k+1) where U = A136214 and
%e A136215 column k of V = column 0 of T^(3k+2) where V = A112333.
%e A136215 This triangle T begins:
%e A136215         1;
%e A136215         1,      1;
%e A136215         4,      2,     1;
%e A136215        28,     12,     3,    1;
%e A136215       280,    112,    24,    4,   1;
%e A136215      3640,   1400,   280,   40,   5,  1;
%e A136215     58240,  21840,  4200,  560,  60,  6, 1;
%e A136215   1106560, 407680, 76440, 9800, 980, 84, 7, 1; ...
%e A136215 Triangle U = A136214 begins:
%e A136215      1;
%e A136215      1,    1;
%e A136215      4,    4,   1;
%e A136215     28,   28,   7,   1;
%e A136215    280,  280,  70,  10,  1;
%e A136215   3640, 3640, 910, 130, 13, 1; ...
%e A136215 with triple factorials A007559 in column 0.
%e A136215 Triangle V = A112333 begins:
%e A136215       1;
%e A136215       2,    1;
%e A136215      10,    5,    1;
%e A136215      80,   40,    8,   1;
%e A136215     880,  440,   88,  11,  1;
%e A136215   12320, 6160, 1232, 154, 14, 1; ...
%e A136215 with triple factorials A008544 in column 0.
%t A136215 T[n_, k_]:= Binomial[n, k]*If[n - k == 0, 1, Product[3*j + 1, {j, 0, n - k - 1}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Jun 10 2018 *)
%o A136215 (PARI) T(n,k)=binomial(n,k)*if(n-k==0,1,prod(j=0,n-k-1,3*j+1))
%Y A136215 Cf. A136216 (matrix square); A007559, A008544; A136212, A136213.
%Y A136215 Cf. A094587.
%K A136215 nonn,tabl
%O A136215 0,4
%A A136215 _Paul D. Hanna_, Feb 07 2008