A123199 -id:A123199 - 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

A123018 Triangle read by rows: row n gives the coefficients of x^k (0 <= k <= n) in the expansion of Sum_{j=0..n} A320508(n,j)x^j(1 - x)^(n - j).

Original entry on oeis.org

1, 1, -2, 1, -2, 2, 1, -2, 1, -1, 1, -2, 0, 2, 0, 1, -2, -1, 5, -4, 0, 1, -2, -2, 8, -7, 2, 1, 1, -2, -3, 11, -9, 0, 3, -2, 1, -2, -4, 14, -10, -6, 12, -6, 2, 1, -2, -5, 17, -10, -16, 27, -15, 3, -1, 1, -2, -6, 20, -9, -30, 47, -24, 0, 4, 0, 1, -2, -7, 23, -7

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 24 2006

The n-th row consists of the coefficients in the expansion of (-x)^n - (1 - x)*(((1 - x - sqrt(1 + 2*x - 3*x^2))/2)^n - ((1 - x + sqrt(1 + 2*x - 3*x^2))/2)^n)/sqrt(1 + 2*x - 3*x^2). - Franck Maminirina Ramaharo, Oct 13 2018

			Triangle begins:
     1;
     1, -2;
     1, -2,  2;
     1, -2,  1, -1;
     1, -2,  0,  2,   0;
     1, -2, -1,  5,  -4,   0;
     1, -2, -2,  8,  -7,   2,  1;
     1, -2, -3, 11,  -9,   0,  3,  -2;
     1, -2, -4, 14, -10,  -6, 12,  -6,   2;
     1, -2, -5, 17, -10, -16, 27, -15,   3, -1;
     1, -2, -6, 20,  -9, -30, 47, -24,   0,  4,  0;
     1, -2, -7, 23,  -7, -48, 71, -28, -18, 22, -8, 0;
     ....

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Row sums: A033999.
Cf. A049310, A168561, A320508.
Cf. A122753, A123019, A123021, A123027, A123199, A123202, A123217, A123221.

P[x_, n_]:= Sum[Binomial[n-k-1, k]*x^k*(1-x)^(n-k), {k, 0, n}];
Table[Coefficient[P[x, n], x, k], {n,0,12}, {k,0,n}]//Flatten (* Franck Maminirina Ramaharo, Oct 14 2018 *)

P(x, n) := sum(binomial(n - k - 1, k)*x^k*(1 - x)^(n - k), k, 0, n)$
create_list(ratcoef(expand(P(x, n)), x, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Oct 14 2018 */

def p(n,x): return sum( binomial(n-j-1, 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 13 2018: (Start)
G.f.: 1/((1 + x*y)*(1 - y + x*y - x*y^2 + x^2*y^2)).
E.g.f.: exp(-x*y) - (exp(y*(1 - x - sqrt(1 + 2*x - 3*x^2))/2) - exp(y*(1 - x + sqrt(1 + 2*x - 3*x^2))/2))*(1 - x)/sqrt(1 + 2*x - 3*x^2). (End)

Edited by N. J. A. Sloane, May 26 2007

Edited by Franck Maminirina Ramaharo, Oct 14 2018

A123027 Triangle of coefficients of (1 - x)^nU(n,-(3x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, -2, 3, 3, -10, 8, -4, 22, -38, 21, 5, -40, 111, -130, 55, -6, 65, -256, 474, -420, 144, 7, -98, 511, -1324, 1836, -1308, 377, -8, 140, -924, 3130, -6020, 6666, -3970, 987, 9, -192, 1554, -6588, 16435, -25088, 23109, -11822, 2584, -10, 255, -2472, 12720, -39430, 77645, -98160, 77378, -34690, 6765

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 24 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A053122(n,j)*x^j*(1 - x)^(n - j).

			Triangle begins:
     1;
    -2,    3;
     3,  -10,    8;
    -4,   22,   38,    21;
     5,  -40,  111,  -130,    55;
    -6,   65, -256,   474,  -420,    144;
     7,  -98,  511, -1324,  1836,  -1308,   377;
    -8,  140, -924,  3130, -6020,   6666, -3970,    987;
     9, -192, 1554, -6588, 16435, -25088, 23109, -11822, 2584;
     ... reformatted and extended. _Franck Maminirina Ramaharo_, Oct 10 2018

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
Wikipedia, Chebyshev polynomials

Cf. A008310, A049310, A053122, A111006.

Cf. A122753, A123018, A123019, A123021, A123199, A123202, A123217, A123221.

b0 = Table[CoefficientList[ChebyshevU[n, x/2 -1], x], {n, 0, 10}];
Table[CoefficientList[Sum[b0[[m+1]][[n+1]]*x^n*(1-x)^(m-n), {n,0,m}], x], {m,0,10}]//Flatten
(* Alternative Adamson Matrix method *)
t[n_, m_] = If[n==m, 2, If[n==m-1 || n==m+1, 1, 0]];
M[d_] := Table[t[n, m], {n, d}, {m, d}];
a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]];
Table[CoefficientList[Sum[a[[m+1]][[n+1]]*x^n*(1-x)^(m-n), {n, 0, m}], x], {m, 0, 10}]//Flatten

A053122(n, k) := if n < k then 0 else ((-1)^(n - k))*binomial(n + k + 1,  2*k + 1)$
P(x, n) := expand(sum(A053122(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
T(n, k) := ratcoef(P(x, n), x, k)$
tabl(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 A053122(n, k): return 0 if (nA053122(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)
Row n = coefficients in the expansion of (1/sqrt((5*x - 4)*x))*(((3*x - 2 + sqrt((5*x - 4)*x))/2)^(n + 1) - ((3*x - 2 - sqrt((5*x - 4)*x))/2)^(n + 1)).
G.f.: 1/(1 + (2 - 3*x)*t + (1 - x)^2*t^2).
E.g.f.: exp(t*(3*x - 2)/2)*(sqrt((5*x - 4)*x)*cosh(t*sqrt((5*x - 4)*x)/2) + (3*x - 2)*sinh(t*sqrt((5*x - 4)*x)/2))/sqrt((5*x - 4)*x).
T(n,1) = (-1)^(n+1)*A006503(n).
T(n,n) = A001906(n+1). (End)

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

A123019 Triangle of coefficients of (1 - x)^n*b(x/(1 - x),n), where b(x,n) is the Morgan-Voyce polynomial related to A085478.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 3, -4, 1, 1, 6, -9, 3, 1, 10, -15, 3, 3, -1, 1, 15, -20, -6, 18, -8, 1, 1, 21, -21, -35, 60, -30, 5, 1, 28, -14, -98, 145, -70, 5, 5, -1, 1, 36, 6, -210, 279, -100, -45, 45, -12, 1, 1, 45, 45, -384, 441, -21, -280, 210, -63, 7, 1, 55, 110

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 24 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A085478(n,j)*x^j*(1 - x)^(n - j).

			Triangle begins:
    1;
    1;
    1,  1,  -1;
    1,  3,  -4,    1;
    1,  6,  -9,    3;
    1, 10, -15,    3,   3,   -1;
    1, 15, -20,   -6,  18,   -8,    1;
    1, 21, -21,  -35,  60,  -30,    5;
    1, 28, -14,  -98, 145,  -70,    5,   5,   -1;
    1, 36,   6, -210, 279, -100,  -45,  45,  -12, 1;
    1, 45,  45, -384, 441,  -21, -280, 210,  -63, 7;
    1, 55, 110, -627, 561,  385, -973, 665, -189, 7, 7, -1;
    ... reformatted and extended. - _Franck Maminirina Ramaharo_, Oct 09 2018

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
Thomas Koshy, Morgan-Voyce Polynomials, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2001, pp. 480-495.
M. N. S. Swamy, Rising Diagonal Polynomials Associated with Morgan-Voyce Polynomials, The Fibonacci Quarterly Vol. 38 (2000), 61-70.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Cf. A078812, A085478.

Cf. A122753, A123018, A123021, A123027, A123199, A123202, A123217, A123221.

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

A085478(n, k) := binomial(n + k, 2*k)$
P(x, n) := expand(sum(A085478(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 09 2018 */

def p(n,x): return sum( binomial(n+j, 2*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

G.f.: (1 - (1 - x)*y)/(1 + (x - 2)*y + (x - 1)^2*y^2). - Vladeta Jovovic, Dec 14 2009
From Franck Maminirina Ramaharo, Oct 10 2018: (Start)
Row n = coefficients in the expansion of (1/(2*sqrt((4 - 3*x)*x)))*((sqrt((4 - 3*x)*x) + x)*((2 - x + sqrt((4 - 3*x)*x))/2)^n + (sqrt((4 - 3*x)*x) - x)*((2 - x - sqrt((4 - 3*x)*x))/2)^n).
E.g.f.: (1/(2*sqrt((4 - 3*x)*x)))*((sqrt((4 - 3*x)*x) + x)*exp(y*(2 - x + sqrt((4 - 3*x)*x))/2) + (sqrt((4 - 3*x)*x) - x)*exp(y*(2 - x - sqrt((4 - 3*x)*x))/2)).
T(n,1) = A000217(n-1). (End)

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 09 2018

A123021 Triangle of coefficients of (1 - x)^n*B(x/(1 - x),n), where B(x,n) is the Morgan-Voyce polynomial related to A078812.

Original entry on oeis.org

1, 2, -1, 3, -2, 4, -2, -2, 1, 5, 0, -9, 6, -1, 6, 5, -24, 18, -4, 7, 14, -49, 36, -4, -4, 1, 8, 28, -84, 50, 20, -30, 10, -1, 9, 48, -126, 36, 115, -120, 45, -6, 10, 75, -168, -48, 358, -335, 120, -6, -6, 1, 11, 110, -198, -264, 847, -714, 175, 84, -63, 14

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 24 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A078812(n,j)*x^j*(1 - x)^(n - j).

			Triangle begins:
    1;
    2,  -1;
    3,  -2;
    4,  -2,   -2,    1;
    5,   0,   -9,    6,  -1;
    6,   5,  -24,   18,  -4;
    7,  14,  -49,   36,  -4,   -4,   1;
    8,  28,  -84,   50,  20,  -30,  10, -1;
    9,  48, -126,   36, 115, -120,  45, -6;
   10,  75, -168,  -48, 358, -335, 120, -6,  -6,  1;
   11, 110, -198, -264, 847, -714, 175, 84, -63, 14, -1;
   ... - _Franck Maminirina Ramaharo_, Oct 09 2018

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
Thomas Koshy, Morgan-Voyce Polynomials, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2001, pp. 480-495.
M. N. S. Swamy, Rising Diagonal Polynomials Associated with Morgan-Voyce Polynomials, The Fibonacci Quarterly Vol. 38 (2000), 61-70.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Cf. A078812, A085478.

Cf. A122753, A123018, A123019, A123027, A123199, A123202, A123217, A123221.

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

t(n, k) := binomial(n + k + 1, n - k)$
P(x, n) := expand(sum(t(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 09 2018 */

def p(n,x): return sum( binomial(n+j+1, 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 09 2018: (Start)
Row n = coefficients in the expansion of (1/sqrt((4 - 3*x)*x))*(((2 - x + sqrt((4 - 3*x)*x))/2)^(n + 1) - ((2 - x - sqrt((4 - 3*x)*x))/2)^(n + 1)).
G.f.: 1/(1 - (2 - x)*y + (1 - x)^2*y^2).
E.g.f.: (1/sqrt((4 - 3*x)*x))*((2 - x + sqrt((4 - 3*x)*x))*exp(y*(2 - x + sqrt((4 - 3*x)*x))/2)/2 - (2 - x - sqrt((4 - 3*x)*x))*exp(y*(2 - x - sqrt((4 - 3*x)*x))/2)/2).
T(n,1) = -A254749(n+1). (End)

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 09 2018

A123202 Triangle of coefficients of n!(1 - x)^nL_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.

Original entry on oeis.org

1, 1, -2, 2, -8, 7, 6, -36, 63, -34, 24, -192, 504, -544, 209, 120, -1200, 4200, -6800, 5225, -1546, 720, -8640, 37800, -81600, 94050, -55656, 13327, 5040, -70560, 370440, -999600, 1536150, -1363572, 653023, -130922, 40320, -645120, 3951360, -12794880

Offset: 0

Roger L. Bagula, Oct 04 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A021009(n,j)*x^j*(1 - x)^(n - j).

			Triangle begins:
       1;
       1,    -2;
       2,    -8,     7;
       6,   -36,    63,    -34;
      24,  -192,   504,   -544,   209;
     120, -1200,  4200,  -6800,  5225,  -1546;
     720, -8640, 37800, -81600, 94050, -55656, 13327;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 13 2018

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, 1972, p. 782.
Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, p. 164, figure 26.1.

G. C. Greubel, Table of n, a(n) for n = 0..5150 (Rows n=0..100 of triangle, flattened; offset corrected by _Georg Fischer_, Jan 31 2019)
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].
Eric Weisstein's World of Mathematics, Laguerre Polynomial

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221.

M := (n,x) -> n!*subs(x=(x/(1-x)),orthopoly[L](n,x))*(1-x)^n:
seq(print(seq(coeff(simplify(M(n,x)),x,k),k=0..n)),n=0..6); # Peter Luschny, Jan 05 2015

w = Table[n!*CoefficientList[LaguerreL[n, x], x], {n, 0, 10}];
v = Table[CoefficientList[Sum[w[[n + 1]][[m + 1]]*x^ m*(1 - x)^(n - m), {m, 0, n}], x], {n, 0, 10}]; Flatten[v]

create_list(ratcoef(n!*(1 - x)^n*laguerre(n, x/(1 - x)), x, k), n, 0, 10, k, 0, n); /* Franck Maminirina Ramaharo, Oct 13 2018 */

row(n) = Vecrev(n!*(1-x)^n*pollaguerre(n, 0, x/(1 - x))); \\ Michel Marcus, Feb 06 2021

T(n, k) = [x^k] (n!*L_n(x)*(1 - x)^n) with L_n(x) the Laguerre polynomial after substituting x by x/(1 - x). - Peter Luschny, Jan 05 2015
From Franck Maminirina Ramaharo, Oct 13 2018: (Start)
G.f.: exp(-x*y/(1 - (1 - x)*y))/(1 - (1 - x)*y).
T(n,1) = A000142(n).
T(n,2) = -A052582(n).
T(n,n) = A002720(n). (End)

Edited by N. J. A. Sloane, Jun 12 2007

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 13 2018

A123217 Triangle read by rows: the n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A123162(n,j)x^j(1 - x)^(n - j).

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 2, 3, -5, 1, 3, 20, -32, 9, 1, 4, 58, -82, 5, 15, 1, 5, 125, -108, -161, 170, -31, 1, 6, 229, 17, -797, 603, 7, -65, 1, 7, 378, 532, -2210, 664, 1468, -968, 129, 1, 8, 580, 1820, -4226, -2846, 8788, -4388, 9, 255, 1, 9, 843, 4440, -5262

Offset: 0

Roger L. Bagula, Oct 04 2006

			Triangle begins:
  1;
  1;
  1, 1,  -1;
  1, 2,   3,   -5;
  1, 3,  20,  -32,     9;
  1, 4,  58,  -82,     5,    15;
  1, 6, 229,   17,  -797,   603,    7,   -65;
  1, 7, 378,  532, -2210,   664, 1468,  -968, 129;
  1, 8, 580, 1820, -4226, -2846, 8788, -4388,   9, 255;
  ... reformatted and extended. - _Franck Maminirina Ramaharo_, Oct 10 2018

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

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123221.

t[n_, k_]= If[k==0, 1, Binomial[2*n-1, 2*k-1]];
p[n_,x_]:= p[n,x]= Sum[t[n,j]*x^j*(1-x)^(n-j), {j,0,n}];
Table[CoefficientList[p[n,x], x], {n, 0, 10}]//Flatten

A123162(n, k) := if n = 0 and k = 0 or k = 0 then 1 else binomial(2*n - 1, 2*k - 1)$
P(x, n) := expand(sum(A123162(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 b(n,k): return 1 if (k==0) else binomial(2*n-1, 2*k-1)
def p(n,x): return sum( b(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)
[T(n) for n in (0..12)] # G. C. Greubel, Jul 15 2021

From Franck Maminirina Ramaharo, Oct 10 2018: (Start)
Row n = coefficients in the expansion of (1-x)^n + x*((1 - 2*sqrt((1-x)*x))^n*(1 - x + sqrt((1-x)*x)) - (1-x - sqrt((1-x)*x))*(1 + 2*sqrt((1-x)*x))^n)/(2*sqrt((1 - x)*x)*(2*x-1)).
G.f.: (1 - (2 - x)*y + (1 - 4*x + 3*x^2)*y^2 - (x - 3*x^2 + 2*x^3)*y^3)/(1 - (3 - x)*y + (3 - 6*x + 4*x^2)*y^2 - (1 - 5*x + 8*x^2 - 4*x^3)*y^3).
E.g.f.: exp((1 - x)*y) + x*((1 - x + sqrt((1 - x)*x))*exp((1 - 2*sqrt((1 - x)*x))*y) - (1 - x - sqrt((1 - x)*x))*exp((1 + 2*sqrt((1 - x)*x))*y))/(2*(2*x - 1)*sqrt((1 - x)*x)) - (1 - 3*x)/(1 - 2*x) + 1. (End)

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 11 2018

A123221 Irregular triangle read by rows: the n-th row consists of the coefficients in the expansion of Sum_{j=0..n(n+1)/2} A008302(n+1,j)x^j*(1 - x)^(n - min(n, j)), where A008302 is the triangle of Mahonian numbers.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 2, 3, 5, 3, 1, 1, 0, 3, 5, 11, 22, 20, 15, 9, 4, 1, 1, 0, 4, 7, 18, 41, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 0, 5, 9, 26, 64, 154, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1, 1, 0, 6, 11, 35, 91, 234, 583

Offset: 0

Roger L. Bagula, Oct 05 2006

			Triangle begins:
    1;
    1;
    1, 0, 1, 1;
    1, 0, 2, 3,  5,  3,  1;
    1, 0, 3, 5, 11, 22, 20,  15,   9,  4,  1;
    1, 0, 4, 7, 18, 41, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1;
    ...

G. C. Greubel, Rows n = 0..20 of the irregular triangle, flattened
Wikipedia, Major index

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217.

M[n_]:= CoefficientList[Product[1-x^j, {j, n}]/(1-x)^n, x];
Table[CoefficientList[Sum[M[n+1][[m+1]]*x^m*(1-x)^(n -Min[n, m]), {m, 0, Binomial[n+1,2]}], x], {n, 0, 10}]//Flatten

A008302(n, k) := ratcoef(ratsimp(product((1 - x^j)/(1 - x), j, 1, n)), x, k)$
P(x, n) := sum(A008302(n + 1, j)*x^j*(1 - x)^(n - min(n, j)), j, 0, n*(n + 1)/2)$
create_list(ratcoef(expand(P(x, n)), x, k), n, 0, 10, k, 0, hipow(P(x, n), x)); /* Franck Maminirina Ramaharo, Oct 14 2018 */

@CachedFunction
def A008302(n,k):
    if (k<0 or k>binomial(n,2)): return 0
    elif (n==1 and k==0): return 1
    else: return A008302(n, k-1) + A008302(n-1, k) - A008302(n-1, k-n)
def p(n,x): return sum( A008302(n+1, j)*x^j*(1-x)^(n-min(n, j)) for j in (0..binomial(n+1,2)) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 16 2021

T(n,k) = A008302(n+1,k) for n + 1 <= k <= n*(n + 1)/2, n > 1. - Franck Maminirina Ramaharo, Oct 14 2018

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 14 2018

A141720 Triangle of coefficients of the polynomials (1 - x)^n*A(n,x/(1 - x)), where A(n,x) are the Eulerian polynomials of A008292.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, -2, 0, 1, 8, -8, 0, 1, 22, -6, -32, 16, 0, 1, 52, 84, -272, 136, 0, 1, 114, 606, -1168, -96, 816, -272, 0, 1, 240, 2832, -2176, -8832, 11904, -3968, 0, 1, 494, 11122, 11072, -83360, 71168, 13312, -31744, 7936, 0, 1, 1004, 39772, 148592, -472760, -17152, 831232, -707584, 176896

Offset: 1

Roger L. Bagula, Sep 11 2008

Row sums are one.
Row n gives the coefficients in the expansion of Sum_{j=1..n} A008292(n,j)*x^j*(1 - x)^(n - j).
The coefficients of the polynomials (1 + x)^n*A(n,x/(1 + x)) are listed in A019538.

			Triangle begins:
  0, 1;
  0, 1;
  0, 1,    2,    -2;
  0, 1,    8,    -8;
  0, 1,   22,    -6,    -32,      16;
  0, 1,   52,    84,   -272,     136;
  0, 1,  114,   606,  -1168,     -96,    816,   -272;
  0, 1,  240,  2832,  -2176,   -8832,  11904,  -3968;
  0, 1,  494, 11122,  11072,  -83360,  71168,  13312,  -31744,   7936;
  0, 1, 1004, 39772, 148592, -472760, -17152, 831232, -707584, 176896;
  ...

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
Eric Weisstein's World of Mathematics, Polylogarithm

Cf. A008292, A019538, A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A144387, A144400, A174128.

R:=PowerSeriesRing(Rationals(), 30);
f:= func< n,x | n eq 0 select 1 else (&+[EulerianNumber(n,j-1)*x^j*(1-x)^(n-j): j in [1..n]]) >;
A141720:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
[A141720(n,k): k in [0..2*Floor((n+1)/2)-1], n in [1..15]]; // G. C. Greubel, Dec 30 2024

CL := p -> PolynomialTools:-CoefficientList(p,x): flatten := seq -> ListTools:-Flatten(seq): flatten([seq(CL(add(A008292(n,j)*x^j*(1-x)^(n-j), j=1..n)), n=1..10)]); # Peter Luschny, Oct 25 2018

Table[CoefficientList[FullSimplify[(1-2x)^(1+n)*PolyLog[-n, x/(1-x)]/(1-x)], x], {n, 1, 10}]//Flatten

def A(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j)^n for j in (0..k))
def p(n,x): return sum( A(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def A141720(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
flatten([A141720(n) for n in range(1,13)]) # G. C. Greubel, Jul 15 2021

Row n is generated by the polynomial (1 - 2*x)^(n+1)*Li(-n, x/(1-x))/(1 - x), where Li(n, z) is the polylogarithm function.
Also generated by Sum_{k=0..n} (eulerian(n,k)*Sum_{l=0..n} (-1)^l*(n - l + 1)*(2 - x)^l*C(l + 1, k)). - Mourad Rahmani (mrahmani(AT)usthb.dz), Jul 22 2010
E.g.f.: (x*exp(2*x*y) - x*exp(y))/(x*exp(y) - (1 - x)*exp(2*x*y)). - Franck Maminirina Ramaharo, Oct 24 2018

Edited by Peter Bala, Jul 04 2012

Edited, and extra term removed by Franck Maminirina Ramaharo, Oct 24 2018

A144387 Triangle read by rows: row n gives the coefficients in the expansion of Sum_{j=0..n} A000040(j+1)x^j(1 - x)^(n - j).

Original entry on oeis.org

2, 2, 1, 2, -1, 4, 2, -3, 5, 3, 2, -5, 8, -2, 8, 2, -7, 13, -10, 10, 5, 2, -9, 20, -23, 20, -5, 12, 2, -11, 29, -43, 43, -25, 17, 7, 2, -13, 40, -72, 86, -68, 42, -10, 16, 2, -15, 53, -112, 158, -154, 110, -52, 26, 13, 2, -17, 68, -165, 270, -312, 264, -162, 78, -13, 18

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

Row sums yield the primes A000040.

			Triangle begins
    2;
    2,   1;
    2,  -1,  4;
    2,  -3,  5,    3;
    2,  -5,  8,   -2,   8;
    2,  -7, 13,  -10,  10,    5;
    2,  -9, 20,  -23,  20,   -5,  12;
    2, -11, 29,  -43,  43,  -25,  17,    7;
    2, -13, 40,  -72,  86,  -68,  42,  -10, 16;
    2, -15, 53, -112, 158, -154, 110,  -52, 26,  13;
    2, -17, 68, -165, 270, -312, 264, -162, 78, -13, 18;
    ...

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

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144400, A174128.

p[x_, n_] = Sum[Prime[k + 1]*x^k*(1 - x)^(n - k), {k, 0, n}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten

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

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 19 2018

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Maxima

Sage

Formula

Extensions

A123018 Triangle read by rows: row n gives the coefficients of x^k (0 <= k <= n) in the expansion of Sum_{j=0..n} A320508(n,j)*x^j*(1 - x)^(n - j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

Sage

Formula

Extensions

A123027 Triangle of coefficients of (1 - x)^n*U(n,-(3*x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

Sage

Formula

Extensions

A123019 Triangle of coefficients of (1 - x)^n*b(x/(1 - x),n), where b(x,n) is the Morgan-Voyce polynomial related to A085478.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

Sage

Formula

Extensions

A123021 Triangle of coefficients of (1 - x)^n*B(x/(1 - x),n), where B(x,n) is the Morgan-Voyce polynomial related to A078812.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

Sage

Formula

Extensions

A123202 Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.

Views

Author

Keywords

Comments

Examples

References

Links

A123018 Triangle read by rows: row n gives the coefficients of x^k (0 <= k <= n) in the expansion of Sum_{j=0..n} A320508(n,j)x^j(1 - x)^(n - j).

A123027 Triangle of coefficients of (1 - x)^nU(n,-(3x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.

A123202 Triangle of coefficients of n!(1 - x)^nL_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.

A123217 Triangle read by rows: the n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A123162(n,j)x^j(1 - x)^(n - j).

A123221 Irregular triangle read by rows: the n-th row consists of the coefficients in the expansion of Sum_{j=0..n(n+1)/2} A008302(n+1,j)x^j*(1 - x)^(n - min(n, j)), where A008302 is the triangle of Mahonian numbers.

A144387 Triangle read by rows: row n gives the coefficients in the expansion of Sum_{j=0..n} A000040(j+1)x^j(1 - x)^(n - j).