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A291451
Triangle read by rows, expansion of e.g.f. exp(x*(exp(z)/3 + 2*exp(-z/2)* cos(z*sqrt(3)/2)/3 - 1)), nonzero coefficients of z.
11
1, 0, 1, 0, 1, 10, 0, 1, 84, 280, 0, 1, 682, 9240, 15400, 0, 1, 5460, 260260, 1401400, 1401400, 0, 1, 43690, 7128576, 99379280, 285885600, 190590400, 0, 1, 349524, 193360720, 6600492080, 42549306800, 76045569600, 36212176000
OFFSET
0,6
EXAMPLE
Triangle starts:
[1]
[0, 1]
[0, 1, 10]
[0, 1, 84, 280]
[0, 1, 682, 9240, 15400]
[0, 1, 5460, 260260, 1401400, 1401400]
[0, 1, 43690, 7128576, 99379280, 285885600, 190590400]
MAPLE
CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
A291451_row := proc(n) exp(x*(exp(z)/3+2*exp(-z/2)*cos(z*sqrt(3)/2)/3-1)):
series(%, z, 66): CL((3*n)!*coeff(series(%, z, 3*(n+1)), z, 3*n), x) end:
for n from 0 to 7 do A291451_row(n) od;
# Alternative:
A291451row := proc(n) local P; P := proc(m, n) option remember;
if n = 0 then 1 else add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
CL(P(3, n), x); seq(%[k+1]/k!, k=0..n) end: # Peter Luschny, Sep 03 2018
MATHEMATICA
P[m_, n_] := P[m, n] = If[n == 0, 1, Sum[Binomial[m*n, m*k]*P[m, n - k]*x, {k, 1, n}]];
row[n_] := Module[{cl = CoefficientList[P[3, n], x]}, Table[cl[[k + 1]]/k!, {k, 0, n}]];
Table[row[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jul 23 2019, after Peter Luschny *)
CROSSREFS
Cf. A048993 (m=1), A156289 (m=2), this seq. (m=3), A291452 (m=4).
Diagonal: A000012 (m=1), A001147 (m=2), A025035 (m=3), A025036 (m=4).
Row sums: A000110 (m=1), A005046 (m=2), A291973 (m=3), A291975 (m=4).
Alternating row sums: A000587 (m=1), A260884 (m=2), A291974 (m=3), A291976 (m=4).
Sequence in context: A030000 A367508 A221809 * A323836 A365893 A062520
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Sep 07 2017
STATUS
approved