A122610 -id:A122610 - cp's OEIS frontend

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

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

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.

			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

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.

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

Cf. A000110, A048993, A088996, A122610.

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

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

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 */

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

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)

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

A122160 Identity matrices minus Steinbach matrices as characteristic polynomials to give a triangular array I[n]-An[i,j]-> P[k,x] P[k,n]->T[n,m).

Original entry on oeis.org

1, 0, -1, -1, -1, 1, -1, 2, 1, -1, -2, 7, -3, -2, 1, -1, 7, -13, 5, 2, -1, -1, 12, -34, 30, -6, -3, 1, 0, 5, -30, 60, -45, 9, 3, -1, 1, 1, -41, 130, -155, 78, -10, -4, 1, 1, -6, -3, 87, -220, 229, -106, 14, 4, -1, 2, -19, 45, 54, -378, 609, -455, 160, -15, -5, 1, 1, -15, 73, -123, -89, 609, -889, 615, -205, 20, 5, -1, 1, -24, 164, -460

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

Remember? 1/(1-x)=Sum[x^n,{n,0,Infinitity}] So to try with the Steinbach field: (I-A[i,j])^(-1)=Sun[A[i,j]^n,{n,0,Infinity}] It doesn't appear it should be finite? But I-A[i,j] is finite--> zero? {{1,0,0}, {{1,1,1}, {{0,-1,-1}, {0,1,0}, {1,1,0}, {-1,0,0}, {0,0,1}} - 1,0,0}}= { -1,0,1}} Matrices: {{0, -1}, {-1, 1}}, {{0, -1, -1}, {-1, 0, 0}, {-1, 0, 1}}, {{0, -1, -1, -1}, {-1, 0, -1, 0}, {-1, -1, 1, 0}, {-1, 0, 0, 1}}, {{0, -1, -1, -1, -1}, {-1, 0, -1, -1, 0}, {-1, -1, 0, 0, 0}, {-1, -1, 0, 1, 0}, {-1, 0, 0, 0, 1}}

			{1},
{0, -1},
{-1, -1, 1},
{-1, 2, 1, -1},
{-2, 7, -3, -2, 1},
{-1, 7, -13, 5, 2, -1},
{-1, 12, -34, 30, -6, -3, 1},
{0, 5, -30, 60, -45, 9,3, -1},
{1, 1, -41, 130, -155, 78, -10, -4, 1}

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

Cf. A065941, A122767, A123218, A066170, A122610.

An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[IdentityMatrix[d] - An[d], x], x], {d, 1, 20}]]; Flatten[%]

I[n]-An[i,j]-> P[k,x] P[k,n]->T[n,m)

cp's OEIS Frontend

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

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