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A187084 Exponential Riordan array (1,4*x+6*x^2+4*x^3+x^4).

%I A187084 #29 Feb 16 2025 08:33:14
%S A187084 4,12,16,24,144,64,24,816,1152,256,0,3360,12480,7680,1024,0,10080,
%T A187084 100800,134400,46080,4096,0,20160,645120,1747200,1182720,258048,16384,
%U A187084 0,20160,3306240,18305280,22364160,9117696,1376256,65536,0,0,13305600,159667200,341591040,235339776,63995904,7077888,262144
%N A187084 Exponential Riordan array (1,4*x+6*x^2+4*x^3+x^4).
%C A187084 The column of index 0 contains a 1 followed by zeros and is not incorporated here.
%C A187084 Also the Bell polynomials of the second kind B(n,k)(4,12,24,24).
%C A187084 If the argument vector is generalized to contain falling powers of a variable x, B(n,k)(4*x^3,12*x,24*x,24) =sum_{j=0..k} binomial(k,j) *sum_{i=j..n-k+j} 6^(i-j) *binomial(j,i-j) *binomial(k-j,n-3*k+3*j-i) *4^(4*k-n-2*j) *x^(4*k-n) *n!/k!.
%H A187084 M. Abbas and S. Bouroubi, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.023">On new identities for Bell's polynomials</a>, Disc. Math 293. (1-3) (2005) 5-10
%H A187084 Vladimir Kruchinin, <a href="http://arxiv.org/abs/1104.5065">Derivation of Bell Polynomials of the Second Kind</a>, arXiv:1104.5065 [math.CO], 2011.
%H A187084 John Riordan, <a href="http://dx.doi.org/10.1090/S0002-9904-1946-08621-8">Derivatives of composite functions</a>, Bull. Am. Math. Soc. 52 (1946) 664
%H A187084 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>, MathWorld
%F A187084 B(n,k) = (n!/k!)*Sum_{j=0..k} binomial(k,j)*( Sum_{i=j..n-k+j} 6^(i-j)*binomial(j,i-j)*binomial(k-j,n-3*k+3*j-i) )*4^(4*k-n-2*j).
%e A187084 Triangle begins:
%e A187084   4;
%e A187084   12,16;
%e A187084   24,144,64;
%e A187084   24,816,1152,256;
%e A187084   0,3360,12480,7680,1024;
%e A187084   0,10080,100800,134400,46080,4096;
%e A187084   0,20160,645120,1747200,1182720,258048,16384;
%e A187084   0,20160,3306240,18305280,22364160,9117696,1376256,65536;
%e A187084   ...
%p A187084 # The function BellMatrix is defined in A264428.
%p A187084 # Adds (1,0,0,0, ..) as column 0.
%p A187084 BellMatrix(n -> `if`(n<4,[4,12,24,24][n+1],0), 9); # _Peter Luschny_, Jan 29 2016
%t A187084 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A187084 B = BellMatrix[Function[n, If[n<4, {4, 12, 24, 24}[[n+1]], 0]], rows = 12];
%t A187084 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A187084 (Maxima)
%o A187084 B(n,k):=n!/k!*sum(binomial(k,j)*sum(6^(i-j)*binomial(j,i-j)*binomial(k-j,n-3*k+3*j-i),i,j,n-k+j)*4^(4*k-n-2*j),j,0,k);
%Y A187084 Cf. A008279 (its row 4), A188138, A188062, A188066.
%K A187084 nonn,tabl
%O A187084 1,1
%A A187084 _Vladimir Kruchinin_, Mar 04 2011