A171694 -id:A171694 - cp's OEIS frontend

A171693 Expansion of g.f.: 2^(1+floor(n/2))n!((1-y)^(n+1)/(1+y))f(x, y, m), where f(x, y, m) = 2^(m+1)exp(2^mt)/((1-yexp(t))(1 + (2^(m+1)-1)exp(2^m*t))), and m = 0.

1, -1, 14, -1, 4, -16, 504, -16, 4, -34, 372, 2178, 35288, 2178, 372, -34, 496, -5888, 65728, 749824, 4185760, 749824, 65728, -5888, 496, -11056, 154912, -767856, 23350656, 230640288, 770603712, 230640288, 23350656, -767856, 154912, -11056

Offset: 0

Roger L. Bagula, Dec 15 2009

			Irregular triangle begins as:
    1;
   -1,    14,    -1;
    4,   -16,   504,    -16,       4;
  -34,   372,  2178,  35288,    2178,    372,   -34;
  496, -5888, 65728, 749824, 4185760, 749824, 65728, -5888, 496;

G. C. Greubel, Rows n = 0..40 of the irregular triangle, flattened

Cf. A060187, A159041, A171692, A171694, A171695.

m= 0;
f[t_, y_, m_]= 2^(m+1)*Exp[2^m*t]/((1-y*Exp[t])*(1+(2^(m+1)-1)*Exp[2^m*t]));
T[n_]:= T[n]= CoefficientList[2^(1+Floor[n/2])*n!*(1-y)^(n+1)/(1 + y)*SeriesCoefficient[Series[f[t, y, m], {t,0,20}], n], y];
Table[T[2*n+1], {n,0,12}]//Flatten (* modified by G. C. Greubel, Mar 30 2022 *)

G.f.: 2^(1+floor(n/2))*n!*((1-y)^(n+1)/(1+y))*f(x, y, m), where f(x, y, m) = 2^(m+1)*exp(2^m*t)/((1-y*exp(t))*(1 + (2^(m+1)-1)*exp(2^m*t))), and m = 0.

Edited by G. C. Greubel, Mar 31 2022

A171695 Expansion of g.f.: 2^(floor((n+1)/2))n!(1-y)^(n+1)f(x, y, m), where f(x, y, m) = 2^(m+1)exp(2^mt)/((1-yexp(t))(1 + (2^(m+1)-1)exp(2^m*t))), and m = 1.

Original entry on oeis.org

1, 1, 1, -1, 6, -1, -1, 7, 25, -7, 10, -44, 152, -20, -2, -26, 198, -292, 1628, -642, 94, -154, 1000, -1954, 6416, 1586, -1400, 266, 1646, -13606, 51774, -75094, 175226, -73890, 15962, -1378, 1000, -3936, -4448, 190432, 37104, 779104, -472160, 133152, -15128

Offset: 0

Roger L. Bagula, Dec 15 2009

			Triangle begins as:
     1;
     1,      1;
    -1,      6,    -1;
    -1,      7,    25,     -7;
    10,    -44,   152,    -20,     -2;
   -26,    198,  -292,   1628,   -642,     94;
  -154,   1000, -1954,   6416,   1586,  -1400,     266;
  1646, -13606, 51774, -75094, 175226, -73890,   15962,  -1378;
  1000,  -3936, -4448, 190432,  37104, 779104, -472160, 133152, -15128;

G. C. Greubel, Rows n = 0..40 of the triangle, flattened

Cf. A060187, A159041, A171692, A171693, A171694.

m= 1;
f[t_, y_, m_]= 2^(m+1)*Exp[2^m*t]/((1-y*Exp[t])*(1+(2^(m+1)-1)*Exp[2^m*t]));
Table[CoefficientList[2^(Floor[(n+1)/2])*n!*(1-y)^(n+1)*SeriesCoefficient[ Series[f[t,y,m], {t,0,20}], n], y], {n,0,12}]//Flatten (* modified by G. C. Greubel, Mar 29 2022 *)

G.f.: 2^(floor((n+1)/2))*n!*(1-y)^(n+1)*f(x, y, m), where f(x, y, m) = 2^(m+1)*exp(2^m*t)/((1-y*exp(t))*(1 + (2^(m+1)-1)*exp(2^m*t))), and m = 1.

Edited by G. C. Greubel, Mar 29 2022

cp's OEIS Frontend

A171693 Expansion of g.f.: 2^(1+floor(n/2))n!((1-y)^(n+1)/(1+y))f(x, y, m), where f(x, y, m) = 2^(m+1)exp(2^mt)/((1-yexp(t))(1 + (2^(m+1)-1)exp(2^m*t))), and m = 0.

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A171695 Expansion of g.f.: 2^(floor((n+1)/2))n!(1-y)^(n+1)f(x, y, m), where f(x, y, m) = 2^(m+1)exp(2^mt)/((1-yexp(t))(1 + (2^(m+1)-1)exp(2^m*t))), and m = 1.

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cp's OEIS Frontend

A171693 Expansion of g.f.: 2^(1+floor(n/2))*n!*((1-y)^(n+1)/(1+y))*f(x, y, m), where f(x, y, m) = 2^(m+1)*exp(2^m*t)/((1-y*exp(t))*(1 + (2^(m+1)-1)*exp(2^m*t))), and m = 0.

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A171695 Expansion of g.f.: 2^(floor((n+1)/2))*n!*(1-y)^(n+1)*f(x, y, m), where f(x, y, m) = 2^(m+1)*exp(2^m*t)/((1-y*exp(t))*(1 + (2^(m+1)-1)*exp(2^m*t))), and m = 1.

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A171693 Expansion of g.f.: 2^(1+floor(n/2))n!((1-y)^(n+1)/(1+y))f(x, y, m), where f(x, y, m) = 2^(m+1)exp(2^mt)/((1-yexp(t))(1 + (2^(m+1)-1)exp(2^m*t))), and m = 0.

A171695 Expansion of g.f.: 2^(floor((n+1)/2))n!(1-y)^(n+1)f(x, y, m), where f(x, y, m) = 2^(m+1)exp(2^mt)/((1-yexp(t))(1 + (2^(m+1)-1)exp(2^m*t))), and m = 1.