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A122753 Triangle of coefficients of (1 - x)^n*B_n(x/(1 - x)), where B_n(x) is the n-th Bell polynomial. 13
1, 0, 1, 0, 1, 0, 1, 1, -1, 0, 1, 4, -5, 1, 0, 1, 11, -14, 1, 2, 0, 1, 26, -24, -29, 36, -9, 0, 1, 57, 1, -244, 281, -104, 9, 0, 1, 120, 225, -1259, 1401, -454, -83, 50, 0, 1, 247, 1268, -5081, 4621, 911, -3422, 1723, -267, 0, 1, 502, 5278, -16981, 5335, 30871 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A048993(n,j)*x^j*(1 - x)^(n - j), where A048993 is the triangle of Stirling numbers of second kind.

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 824-825.

LINKS

Table of n, a(n) for n=0..60.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Eric Weisstein's World of Mathematics, Bell Polynomial

Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind

FORMULA

From Franck Maminirina Ramaharo, Oct 10 2018: (Start)

E.g.f.: exp((x/(1 - x))*(exp((1 - x)*y) - 1).

T(n,1) = A000295(n-1). (End)

EXAMPLE

Triangle begins:

    1;

    0, 1;

    0, 1;

    0, 1,   1,   -1;

    0, 1,   4,   -5,     1;

    0, 1,  11,  -14,     1,    2;

    0, 1,  26,  -24,   -29,   36,   -9;

    0, 1,  57,    1,  -244,  281, -104,     9;

    0, 1, 120,  225, -1259, 1401, -454,   -83,   50;

    0, 1, 247, 1268, -5081, 4621,  911, -3422, 1723, -267;

    ... reformatted and extended. - Franck Maminirina Ramaharo, Oct 10 2018

MATHEMATICA

a = Table[CoefficientList[Sum[StirlingS2[m, n]*x^n*(1 - x)^(m - n), {n, 0, m}], x], {m, 0, 10}]; Flatten[a]

PROG

(Maxima)

P(x, n) := expand(sum(stirling2(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$

T(n, k) := ratcoef(P(x, n), x, k)$

tabf(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* Franck Maminirina Ramaharo, Oct 10 2018 */

CROSSREFS

Cf. A000110, A048993, A088996, A122610.

Cf. A123018, A123019, A123021, A123027, A123199, A123202, A123202, A123217, A123221.

Sequence in context: A178219 A322232 A232397 * A016714 A211799 A113950

Adjacent sequences:  A122750 A122751 A122752 * A122754 A122755 A122756

KEYWORD

sign,tabf

AUTHOR

Roger L. Bagula and Gary W. Adamson, Sep 21 2006

EXTENSIONS

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018

STATUS

approved

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Last modified December 14 17:55 EST 2019. Contains 329979 sequences. (Running on oeis4.)