A187084 Exponential Riordan array (1,4*x+6*x^2+4*x^3+x^4).
4, 12, 16, 24, 144, 64, 24, 816, 1152, 256, 0, 3360, 12480, 7680, 1024, 0, 10080, 100800, 134400, 46080, 4096, 0, 20160, 645120, 1747200, 1182720, 258048, 16384, 0, 20160, 3306240, 18305280, 22364160, 9117696, 1376256, 65536, 0, 0, 13305600, 159667200, 341591040, 235339776, 63995904, 7077888, 262144
Offset: 1
Examples
Triangle begins: 4; 12,16; 24,144,64; 24,816,1152,256; 0,3360,12480,7680,1024; 0,10080,100800,134400,46080,4096; 0,20160,645120,1747200,1182720,258048,16384; 0,20160,3306240,18305280,22364160,9117696,1376256,65536; ...
Links
- M. Abbas and S. Bouroubi, On new identities for Bell's polynomials, Disc. Math 293. (1-3) (2005) 5-10
- Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2011.
- John Riordan, Derivatives of composite functions, Bull. Am. Math. Soc. 52 (1946) 664
- Eric W. Weisstein, Bell Polynomial, MathWorld
Programs
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Maple
# The function BellMatrix is defined in A264428. # Adds (1,0,0,0, ..) as column 0. BellMatrix(n -> `if`(n<4,[4,12,24,24][n+1],0), 9); # Peter Luschny, Jan 29 2016
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Mathematica
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]]; B = BellMatrix[Function[n, If[n<4, {4, 12, 24, 24}[[n+1]], 0]], rows = 12]; Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *) -
Maxima
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);
Formula
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).
Comments