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A187084 Exponential Riordan array (1,4*x+6*x^2+4*x^3+x^4). 0
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, 0, 0, 39916800 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The column of index 0 contains a 1 followed by zeros and is not incorporated here.
Also the Bell polynomials of the second kind B(n,k)(4,12,24,24).
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!.
LINKS
M. Abbas, 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
John Riordan, Derivatives of composite functions, Bull. Am. Math. Soc. 52 (1946) 664
Eric W. Weisstein, Bell Polynomial, MathWorld
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).
EXAMPLE
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;
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
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 *)
PROG
(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);
CROSSREFS
Cf. A008279 (its row 4), A188138, A188062, A188066.
Sequence in context: A057962 A186303 A073687 * A090818 A075191 A320922
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Mar 04 2011
STATUS
approved

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Last modified February 29 11:59 EST 2024. Contains 370425 sequences. (Running on oeis4.)