A122753 Triangle of coefficients of (1 - x)^n*B_n(x/(1 - x)), where B_n(x) is the n-th Bell polynomial.

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

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

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

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

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

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 */
(Sage)
def p(n, x): return sum( stirling_number2(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 15 2021

Crossrefs

Cf. A000110, A048993, A088996, A122610.

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

Sequence in context: A322232 A232397 A360220 * 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