A088996 -id:A088996 - 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

A142070 Triangle T(n,k) read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (i+1)*x-i in row n>=0 and column 0<=k<=n.

Original entry on oeis.org

1, -1, 2, 2, -7, 6, -6, 29, -46, 24, 24, -146, 329, -326, 120, -120, 874, -2521, 3604, -2556, 720, 720, -6084, 21244, -39271, 40564, -22212, 5040, -5040, 48348, -197380, 444849, -598116, 479996, -212976, 40320, 40320, -432144, 2014172, -5335212, 8788569, -9223012, 6023772, -2239344, 362880

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 15 2008

This is essentially a signed version of A088996. - Peter Bala, Jan 09 2017

			Triangle begins as:
      1;
     -1,       2;
      2,      -7,       6;
     -6,      29,     -46,       24;
     24,    -146,     329,     -326,     120;
   -120,     874,   -2521,     3604,   -2556,      720;
    720,   -6084,   21244,   -39271,   40564,   -22212,    5040;
  -5040,   48348, -197380,   444849, -598116,   479996, -212976,    40320;
  40320, -432144, 2014172, -5335212, 8788569, -9223012, 6023772, -2239344, 362880;

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

Cf. A048994, A088996.

A142070:= func< n,k | (-1)^(n-k)*(&+[(-1)^j*Binomial(j,n-k)*StirlingFirst(n+1,n-j+1): j in [0..n]]) >;
[A142070(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 24 2022

A142070 := proc(n,k)
    local x,i ;
    mul( (i+1)*x-i,i=1..n) ;
    expand(%) ;
    coeff(%,x,k) ;
end proc:

(* First program *)
p[x_, n_]:= Product[(i+1)*x - i, {i, n}];
Table[CoefficientList[p[x, n], x], {n,0,10}]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k]= Sum[(-1)^j*Binomial[j+n-k, n-k]*StirlingS1[n+1,k-j+1], {j, 0, k}];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 24 2022 *)

row(n) = Vecrev(prod(j=1, n, (1+j)*x - j)); \\ Michel Marcus, Feb 24 2022

def A142070(n,k): return (-1)^(n-k)*sum(binomial(j+n-k, n-k)*stirling_number1(n+1, k-j+1) for j in (0..k))
flatten([[A142070(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 24 2022

T(n, k) = [x^k]( Product_{j=1..n} ((1+j)*x - j) ).
Sum_{k=0..n} T(n, k) = 1.
From G. C. Greubel, Feb 24 2022: (Start)
T(n, k) = (-1)^(n-k) * Sum_{j=0..n} (-1)^j*binomial(j,n-k)*Stirling1(n+1, n-j+1).
T(n, k) = Sum_{j=0..k} (-1)^j*binomial(j+n-k,n-k)*Stirling1(n+1, k-j+1).
T(n, 0) = (-1)^n * n!.
T(n, n) = (n+1)!. (End)

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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