A159041 -id:A159041 - cp's OEIS frontend

A171692 Triangle read by rows: absolute values of odd-numbered rows of A159041.

1, 1, 10, 1, 1, 56, 246, 56, 1, 1, 246, 4047, 11572, 4047, 246, 1, 1, 1012, 46828, 408364, 901990, 408364, 46828, 1012, 1, 1, 4082, 474189, 9713496, 56604978, 105907308, 56604978, 9713496, 474189, 4082, 1, 1, 16368, 4520946, 193889840, 2377852335, 10465410528, 17505765564, 10465410528, 2377852335, 193889840, 4520946, 16368, 1

Offset: 0

Roger L. Bagula, Dec 15 2009

			Irregular triangle begins as:
  1;
  1,   10,     1;
  1,   56,   246,     56,      1;
  1,  246,  4047,  11572,   4047,    246,     1;
  1, 1012, 46828, 408364, 901990, 408364, 46828, 1012, 1;

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

Cf. A008292, A060187, A159041.

(* First program *)
f[x_, y_, m_]:= 2^(m+1)*Exp[2^m*x]/((1 -y*Exp[x])*(1 +(2^(m+1) -1)*Exp[2^m*x]));
Table[CoefficientList[SeriesCoefficient[Series[((1-y)^(n+1)/(2*y))*n!*f[x, y, 0], {x,0,30}], n], y], {n, 2, 20, 2}]//Flatten (* modified by G. C. Greubel, Mar 18 2022 *)
(* Second program *)
A008292[n_, k_]:= Sum[(-1)^j*(k-j)^n*Binomial[n+1, j], {j,0,k}];
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], T[n, k-1] + (-1)^k*A008292[n+2, k+1], T[n, n-k] ]]; (* T = A159041 *)
A171692[n_, k_]:= Abs[T[2*n, k]];
Table[A171692[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Mar 18 2022 *)

def A008292(n,k): return sum( (-1)^j*(k-j)^n*binomial(n+1,j) for j in (0..k) )
@CachedFunction
def A159041(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return A159041(n,k-1) + (-1)^k*A008292(n+2,k+1)
    else: return A159041(n,n-k)
def A171692(n,k): return abs( A159041(2*n, k) )
flatten([[A171692(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

T(n, k) = coefficients of (g(x, y)), where g(x, y) = n! * ((1-y)^(n+1)/(2*y)) * f(x, y, 0), with f(x, y, m) = 2^(m+1)*exp(2^m*x)/((1 -y*exp(x))*(1 +(2^(m+1) -1)*exp(2^m*x))).
From G. C. Greubel, Mar 18 2022: (Start)
T(n, k) = abs( A159041(2*n, k) ).
T(n, n-k) = T(n, k). (End)

Edited by N. J. A. Sloane, May 10 2013
More terms from Jean-François Alcover, Feb 14 2014
Edited by G. C. Greubel, Mar 18 2022

A225434 Apply the triangle-to-triangle transformation described in the Comments in A159041 to the triangle in A142459.

Original entry on oeis.org

1, 1, 1, 1, -58, 1, 1, -307, -307, 1, 1, -1556, 12006, -1556, 1, 1, -7805, 140722, 140722, -7805, 1, 1, -39054, 1461615, -5647300, 1461615, -39054, 1, 1, -195303, 14287093, -109642851, -109642851, 14287093, -195303, 1, 1, -976552, 135028828, -1838120344, 4873361350, -1838120344, 135028828, -976552, 1

Offset: 0

Roger L. Bagula, May 07 2013

			The triangle begins:
  1;
  1,       1;
  1,     -58,        1;
  1,    -307,     -307,          1;
  1,   -1556,    12006,      -1556,          1;
  1,   -7805,   140722,     140722,      -7805,        1;
  1,  -39054,  1461615,   -5647300,    1461615,   -39054,       1;
  1, -195303, 14287093, -109642851, -109642851, 14287093, -195303, 1;

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

Cf. A142459, A159041.

Maple
```
See A159041.
```

(* First program *)
t[n_, k_, m_]:= t[n, k, m]= If[k==0 || k==n, 1, (m*(n+1)-m*(k+1)+1)*t[n-1,k-1,m] + (m*(k+1)-(m-1))*t[n-1,k,m] ]; (* t(n,k,4)=A142459 *)
p[x_, n_]:= p[x, n]= Sum[x^i*If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i<=Floor[n/2], (-1)^i*t[n,i,4], (-1)^(n-i+1)*t[n,i,4]]], {i,0,n}]/(1-x);
Flatten[Table[CoefficientList[p[x, n], x], {n,0,12}]]
(* Second program *)
t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1, (m*(n+1)-m*(k+1)+1)*t[n-1,k-1,m] + (m*(k+1)-(m-1))*t[n-1,k,m]];
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], T[n, k-1] + (-1)^k*t[n+2,k+1,4], T[n, n-k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 19 2022 *)

@CachedFunction
def T(n, k, m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
def A142459(n,k): return T(n,k,4)
@CachedFunction
def A225434(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return A225434(n,k-1) + (-1)^k*A142459(n+2,k+1)
    else: return A225434(n,n-k)
flatten([[A225434(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 19 2022

A triangle of polynomial coefficients: p(x,n) = Sum_{i=0..n} ( x^i * if(i = floor(n/2) and (n mod 2) = 0, 0, if(i <= floor(n/2), (-1)^i*A142459(n+1, i+1), (-1)^(n-i+1)*A142459(n+1, i+1) ) )/(1-x).

T(n, k) = T(n,k-1) + (-1)^k*A142459(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1. - G. C. Greubel, Mar 19 2022

Edited by N. J. A. Sloane, May 11 2013

A225483 Triangle T(n, k) = Sum_{j=0..k} (-1)^(k-j)A159041(2n+1, j), read by rows.

Original entry on oeis.org

1, 1, -26, 1, 1, -120, 1192, -120, 1, 1, -502, 14609, -88736, 14609, -502, 1, 1, -2036, 152638, -2205524, 9890752, -2205524, 152638, -2036, 1, 1, -8178, 1479727, -45541628, 424761262, -1551163136, 424761262, -45541628, 1479727, -8178, 1

Offset: 0

Roger L. Bagula, May 08 2013

			The triangle begins:
  1;
  1,   -26,      1;
  1,  -120,   1192,     -120,       1;
  1,  -502,  14609,   -88736,   14609,     -502,      1;
  1, -2036, 152638, -2205524, 9890752, -2205524, 152638, -2036, 1;

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

Cf. A008292, A060187, A158041.

(* First program *)
Needs["Combinatorica`"];
p[n_, x_]:= p[n,x]= Sum[If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i<=Floor[n/2], (-1)^i*Eulerian[n+1, i]*x^i, (-1)^(n-i+1)*Eulerian[n+1, i]*x^i]], {i,0,n}]/(1- x^2);
Table[CoefficientList[p[x, 2*n], x], {n,0,10}]//Flatten
(* Second program *)
A008292[n_, k_]:= A008292[n, k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];
f[n_, k_]:= f[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], f[n, k-1] + (-1)^k*A008292[n+2, k+1], f[n, n-k]]]; (* f = A159041 *)
T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j)*f[2*n+1,j], {j,0,k}];
Table[T[n, k], {n,0,10}, {k,0,2*n}]//Flatten (* G. C. Greubel, Mar 29 2022 *)

def A008292(n, k): return sum( (-1)^j*(k-j)^n*binomial(n+1, j) for j in (0..k) )
@CachedFunction
def f(n, k): # A159041
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return f(n, k-1) + (-1)^k*A008292(n+2, k+1)
    else: return f(n, n-k)
def A225483(n,k): return sum( (-1)^(k-j)*f(2*n+1,j) for j in (0..k) )
flatten([[A225483(n, k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 29 2022

T(n, k) = [x^k]( A159041(x,n)/(x+1) ).
From G. C. Greubel, Mar 29 2022: (Start)
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j).
T(n, 2*n-k) = T(n, k). (End)

Edited by G. C. Greubel, Mar 29 2022

A225356 Triangle T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -22, 1, 1, -75, -75, 1, 1, -236, 1446, -236, 1, 1, -721, 9822, 9822, -721, 1, 1, -2178, 58479, -201244, 58479, -2178, 1, 1, -6551, 325061, -2160227, -2160227, 325061, -6551, 1, 1, -19672, 1736668, -19971304, 49441990, -19971304, 1736668, -19672, 1

Offset: 0

Roger L. Bagula, May 07 2013

			The triangle begins:
  1;
  1,      1;
  1,    -22,       1;
  1,    -75,     -75,         1;
  1,   -236,    1446,      -236,        1;
  1,   -721,    9822,      9822,     -721,         1;
  1,  -2178,   58479,   -201244,    58479,     -2178,       1;
  1,  -6551,  325061,  -2160227, -2160227,    325061,   -6551,      1;
  1, -19672, 1736668, -19971304, 49441990, -19971304, 1736668, -19672, 1;

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

Cf A007318, A060187, A159041.

(* First program *)
q[x_, n_]= (1-x)^(n+1)*Sum[(2*m+1)^n*x^m, {m, 0, Infinity}];
t[n_, m_]:= t[n, m]= Table[CoefficientList[q[x, k], x], {k,0,15}][[n+1, m+1]];
p[x_, n_]:= p[x, n]= Sum[x^i*If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i <= Floor[n/2], (-1)^i*t[n, i], (-1)^(n-i+1)*t[n, i]]], {i,0,n}]/(1-x);
Flatten[Table[CoefficientList[p[x, n], x], {n,10}]]
(* Second Program *)
A060187[n_, k_]:= Sum[(-1)^(k-i)*Binomial[n, k-i]*(2*i-1)^(n-1), {i,k}];
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], T[n, k-1] +(-1)^k*A060187[n+2, k+1], T[n, n-k] ]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2022 *)

def A060187(n,k): return sum( (-1)^(k-j)*(2*j-1)^(n-1)*binomial(n, k-j) for j in (1..k) )
@CachedFunction
def A225356(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return A225356(n,k-1) + (-1)^k*A060187(n+2,k+1)
    else: return A225356(n,n-k)
flatten([[A225356(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1.

Edited by N. J. A. Sloane, May 11 2013

Edited by G. C. Greubel, Mar 18 2022

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.

Original entry on oeis.org

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

A171694 Expansion of g.f.: 4^nn!(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 = -2.

Original entry on oeis.org

1, 2, 2, 6, 20, 6, 26, 154, 190, 14, 150, 1160, 3428, 1352, 54, 1082, 9174, 50404, 51724, 10434, 62, 9366, 78476, 683962, 1376232, 734122, 65996, 966, 94586, 735410, 9096210, 30488714, 32703374, 8931318, 530534, -4786, 1091670, 7562000, 122859048, 611454960, 1132022084, 653476464, 111158184, 2715536, 71574

Offset: 0

Roger L. Bagula, Dec 15 2009

			Triangle begins as:
      1;
      2,      2;
      6,     20,       6;
     26,    154,     190,       14;
    150,   1160,    3428,     1352,       54;
   1082,   9174,   50404,    51724,    10434,      62;
   9366,  78476,  683962,  1376232,   734122,   65996,    966;
  94586, 735410, 9096210, 30488714, 32703374, 8931318, 530534, -4786;

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

Cf. A060187, A159041, A171692, A171693.

m= -2;
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[4^n*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.: 4^n*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 = -2.

Edited by G. C. Greubel, Mar 29 2022

A225433 Triangle T(n, k) = T(n, k-1) + (-1)^k*A142458(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -38, 1, 1, -165, -165, 1, 1, -676, 4806, -676, 1, 1, -2723, 44452, 44452, -2723, 1, 1, -10914, 362895, -1346780, 362895, -10914, 1, 1, -43681, 2780367, -20297327, -20297327, 2780367, -43681, 1, 1, -174752, 20554588, -263879264, 683233990, -263879264, 20554588, -174752, 1

Offset: 0

Roger L. Bagula, May 07 2013

			The triangle begins:
  1;
  1,      1;
  1,    -38,       1;
  1,   -165,    -165,         1;
  1,   -676,    4806,      -676,         1;
  1,  -2723,   44452,     44452,     -2723,       1;
  1, -10914,  362895,  -1346780,    362895,  -10914,      1;
  1, -43681, 2780367, -20297327, -20297327, 2780367, -43681, 1;

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

Cf. A007318, A060187, A142458, A159041, A225356.

Maple
```
See Maple program in A159041.
```

(* First program *)
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n+1) -m*(k+1) +1)*T[n-1, k- 1, m] + (m*(k+1) -(m-1))*T[n-1, k, m] ];
p[x_, n_]:= p[x, n]= Sum[x^i*If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i<= Floor[n/2], (-1)^i*T[n,i,3], -(-1)^(n-i)*T[n,i,3]]], {i,0,n}]/(1-x);
Flatten[Table[CoefficientList[p[x, n], x], {n,0,12}]]
(* Second program *)
T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k-m+1)*T[n-1, k, m]];
A142458[n_, k_]:= T[n,k,3];
A225433[n_, k_]:= A225433[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], A225433[n, k-1] +(-1)^k*A142458[n+2, k+1], A225433[n, n-k]]];
Table[A225433[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 19 2022 *)

@CachedFunction
def T(n, k, m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
def A142458(n,k): return T(n,k,3)
@CachedFunction
def A225433(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return A225433(n,k-1) + (-1)^k*A142458(n+2,k+1)
    else: return A225433(n,n-k)
flatten([[A225433(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 19 2022

From G. C. Greubel, Mar 19 2022: (Start)
T(n, k) = T(n, k-1) + (-1)^k*A142458(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1.
T(n, n-k) = T(n, k). (End)

Edited by N. J. A. Sloane, May 11 2013

Edited by G. C. Greubel, Mar 19 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

A225532 Triangle T(n, k) = abs(A225483(n/2, k)) if (n mod 2 = 0), otherwise abs(A225482((n-1)/2, k) - A225483((n-1)/2, k-1)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 26, 1, 1, 27, 27, 1, 1, 120, 1192, 120, 1, 1, 121, 1312, 1312, 121, 1, 1, 502, 14609, 88736, 14609, 502, 1, 1, 503, 15111, 103345, 103345, 15111, 503, 1, 1, 2036, 152638, 2205524, 9890752, 2205524, 152638, 2036, 1, 1, 2037, 154674, 2358162, 12096276, 12096276, 2358162, 154674, 2037, 1

Offset: 0

Roger L. Bagula, May 09 2013

			Triangle begins:
  1;
  1,   1;
  1,  26,     1;
  1,  27,    27,      1;
  1, 120,  1192,    120,      1;
  1, 121,  1312,   1312,    121,     1;
  1, 502, 14609,  88736,  14609,   502,   1;
  1, 503, 15111, 103345, 103345, 15111, 503, 1;

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

Cf. A008292, A159041, A171692, A204621, A225483.

(* First program *)
Needs["Combinatorica`"];
p[n_, x_]:= p[n,x]= Sum[If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i<=Floor[n/2], (-1)^i*Eulerian[n+1,i]*x^i, (-1)^(n-i+1)*Eulerian[n+1,i]*x^i]], {i,0,n}]/(1 - x^2);
q[n_, x_]= If[Mod[n,2]==0, (1-x)*p[n/2,x], p[(n+1)/2,x]];
Table[Abs[CoefficientList[q[(4*n +(-1)^n +5)/2, x], x]], {n,0,12}]//Flatten (* modified by G. C. Greubel, Mar 29 2022 *)
(* Second program *)
A008292[n_, k_]:= A008292[n, k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];
f[n_, k_]:= f[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], f[n,k-1] + (-1)^k*A008292[n+2,k+1], f[n,n-k]]]; (* f=A159041 *)
A225483[n_, k_]:= Sum[(-1)^(k-j)*f[2*n+1,j], {j,0,k}];
T[n_, k_]:= If[Mod[n,2]==0, A225483[n/2, k], A225483[(n-1)/2, k] - A225483[(n - 1)/2, k-1] ]//Abs;
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 29 2022 *)

def A008292(n, k): return sum( (-1)^j*(k-j)^n*binomial(n+1, j) for j in (0..k) )
@CachedFunction
def f(n, k): # A159041
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return f(n, k-1) + (-1)^k*A008292(n+2, k+1)
    else: return f(n, n-k)
def A225483(n,k): return sum( (-1)^(k-j)*f(2*n+1,j) for j in (0..k) )
@CachedFunction
def A225532(n,k):
    if (n%2==0): return abs(A225483(n/2, k))
    else: return abs( A225483((n-1)/2, k) - A225483((n-1)/2, k-1) )
flatten([[A225532(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 29 2022

From G. C. Greubel, Mar 29 2022: (Start)
T(n, k) = abs(A225483(n/2, k)) if (n mod 2 = 0), otherwise abs(A225482((n-1)/2, k) - A225483((n-1)/2, k-1)).
T(n, n-k) = T(n, k). (End)

Edited by G. C. Greubel, Mar 29 2022

A347816 Prime numbers p such that both 15 and 85 are quadratic nonresidue (mod p).

Original entry on oeis.org

13, 29, 31, 41, 47, 79, 83, 139, 157, 199, 211, 263, 269, 373, 379, 383, 401, 433, 439, 443, 449, 457, 467, 499, 521, 563, 571, 577, 587, 613, 619, 641, 647, 691, 733, 751, 757, 809, 811, 821, 863, 881, 929, 937, 941, 991, 1033, 1049, 1051, 1061

Offset: 1

Sela Fried, Sep 15 2021

nonn

Primes p such that E_6(x)/(x + 1) is irreducible (mod p) where E_6(x) is the Eulerian polynomial and E_6(x)/(x + 1) = x^4 + 56x^3 + 246x^2 + 56x + 1. (See A159041.)
The sequence is infinite.
It is the intersection of A038888 and A038972.

A. J. J. Heidrich, On the factorization of Eulerian polynomials, Journal of Number Theory, 18(2):157-168, 1984.

Cf. A159041, A008292, A173018, A038888, A038972, A347815.

alias(ls = NumberTheory:-LegendreSymbol):
isA347816 := k -> isprime(k) and ls(15, k) = -1 and ls(85, k) = -1:
A347816List := upto -> select(isA347816, [`$`(3..upto)]):
A347816List(1061); # Peter Luschny, Sep 16 2021

Select[Prime@Range[180], JacobiSymbol[15, #] == -1 && JacobiSymbol[85,#]==-1 &] (* Stefano Spezia, Sep 16 2021 *)

isok(p) = isprime(p) && (kronecker(15,p)==-1) && (kronecker(85,p)==-1); \\ Michel Marcus, Sep 16 2021

from sympy.ntheory import legendre_symbol, primerange
A347816_list = [p for p in primerange(3,10**5) if legendre_symbol(15,p) == legendre_symbol(85,p) == -1] # Chai Wah Wu, Sep 16 2021

cp's OEIS Frontend

A171692 Triangle read by rows: absolute values of odd-numbered rows of A159041.

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A225434 Apply the triangle-to-triangle transformation described in the Comments in A159041 to the triangle in A142459.

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A225483 Triangle T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j), read by rows.

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A225356 Triangle T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

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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^m*t)/((1-y*exp(t))*(1 + (2^(m+1)-1)*exp(2^m*t))), and m = 0.

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A171694 Expansion of g.f.: 4^n*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 = -2.

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A225433 Triangle T(n, k) = T(n, k-1) + (-1)^k*A142458(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

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A225483 Triangle T(n, k) = Sum_{j=0..k} (-1)^(k-j)A159041(2n+1, j), read by rows.

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.

A171694 Expansion of g.f.: 4^nn!(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 = -2.

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.