cp's OEIS Frontend

A136590 Triangle of trinomial logarithmic coefficients: A027907(n,k) = Sum_{i=0..k} T(k,i)*n^i/k!.

%I A136590 #17 Mar 27 2020 08:47:20
%S A136590 1,0,1,0,1,1,0,-4,3,1,0,6,-13,6,1,0,24,-10,-25,10,1,0,-240,394,-135,
%T A136590 -35,15,1,0,720,-2016,1834,-525,-35,21,1,0,5040,-5076,-3668,5089,
%U A136590 -1400,-14,28,1,0,-80640,170064,-110692,14364,9849,-3024,42,36,1,0,362880,-1155024,1339020,-672400,118125,12873,-5670,150,45,1
%N A136590 Triangle of trinomial logarithmic coefficients: A027907(n,k) = Sum_{i=0..k} T(k,i)*n^i/k!.
%C A136590 A027907 is the triangle of trinomial coefficients.
%C A136590 The Bell transform of A136591(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%F A136590 E.g.f. of column k = log(1 + x + x^2)^k / k! for k>=0.
%F A136590 Central trinomial coefficients: A002426(n) = Sum_{k=0..n} T(n,k)*n^k/n!.
%e A136590 Triangle begins:
%e A136590 1;
%e A136590 0, 1;
%e A136590 0, 1, 1;
%e A136590 0, -4, 3, 1;
%e A136590 0, 6, -13, 6, 1;
%e A136590 0, 24, -10, -25, 10, 1;
%e A136590 0, -240, 394, -135, -35, 15, 1;
%e A136590 0, 720, -2016, 1834, -525, -35, 21, 1;
%e A136590 0, 5040, -5076, -3668, 5089, -1400, -14, 28, 1;
%e A136590 0, -80640, 170064, -110692, 14364, 9849, -3024, 42, 36, 1;
%e A136590 0, 362880, -1155024, 1339020, -672400, 118125, 12873, -5670, 150, 45, 1; ...
%e A136590 Trinomial coefficients can be calculated as illustrated by:
%e A136590 A027907(4,3) = (T(3,0)*4^0 + T(3,1)*4^1 + T(3,2)*4^2 + T(3,3)*4^3)/3! =
%e A136590 (0 - 4*4 + 3*4^2 + 1*4^3)/3! = 96/6 = 16.
%p A136590 # The function BellMatrix is defined in A264428.
%p A136590 BellMatrix(n -> n!*(modp(n+1,3)-modp(n,3)), 9); # _Peter Luschny_, Jan 27 2016
%t A136590 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A136590 rows = 11;
%t A136590 M = BellMatrix[#!*(Mod[# + 1, 3] - Mod[#, 3])&, rows];
%t A136590 Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%o A136590 (PARI) {T(n,k)=n!/k!*polcoeff(log(1+x+x^2 +x*O(x^n))^k,n)}
%o A136590 for(n=0,10, for(k=0,n, print1( T(n,k),", "));print(""))
%o A136590 (Sage) # uses[bell_matrix from A264428]
%o A136590 bell_matrix(lambda n: A136591(n+1), 10) # _Peter Luschny_, Jan 18 2016
%Y A136590 Cf. columns: A136591, A136592, A136593; A136594 (unsigned row sums); A136595 (matrix inverse); A027907, A002426.
%K A136590 sign,tabl
%O A136590 0,8
%A A136590 _Paul D. Hanna_, Jan 10 2008