A171692 -id:A171692 - cp's OEIS frontend

A159041 Triangle read by rows: row n (n>=0) gives the coefficients of the polynomial p(n,x) of degree n defined in comments.

1, 1, 1, 1, -10, 1, 1, -25, -25, 1, 1, -56, 246, -56, 1, 1, -119, 1072, 1072, -119, 1, 1, -246, 4047, -11572, 4047, -246, 1, 1, -501, 14107, -74127, -74127, 14107, -501, 1, 1, -1012, 46828, -408364, 901990, -408364, 46828, -1012, 1, 1, -2035, 150602, -2052886, 7685228, 7685228, -2052886, 150602, -2035, 1

Offset: 0

Roger L. Bagula, Apr 03 2009

Let E(n,k) (1 <= k <= n) denote the Eulerian numbers as defined in A008292. Then we define polynomials p(n,x) for n >= 0 as follows.
p(n,x) = (1/(1-x)) * ( Sum_{k=0..floor(n/2)} (-1)^k*E(n+2,k+1)*x^k + Sum_{k=ceiling((n+2)/2)..n+1} (-1)^(n+k)*E(n+2,k+1)*x^k ).
For example,
p(0,x) = (1-x)/(1-x) = 1,
p(1,x) = (1-x^2)/(1-x) = 1 + x,
p(2,x) = (1 - 11*x + 11*x^2 - x^3)/(1-x) = 1 - 10*x + x^2,
p(3,x) = (1 - 26*x + 26*x^3 - x^4)/(1-x) = 1 - 25*x - 25*x^2 + x^3,
p(4,x) = (1 - 57*x + 302*x^2 - 302*x^3 + 57*x^3 + x^5)/(1-x)
       = 1 - 56*x + 246*x^2 - 56*x^3 + x^4.
More generally, there is a triangle-to-triangle transformation U -> T defined as follows.
Let U(n,k) (1 <= k <= n) be a triangle of nonnegative numbers in which the rows are symmetric about the middle. Define polynomials p(n,x) for n >= 0 by
p(n,x) = (1/(1-x)) * ( Sum_{k=0..floor(n/2)} (-1)^k*U(n+2,k+1)*x^k + Sum_{k=ceiling((n+2)/2)..n+1} (-1)^(n+k)*U(n+2,k+1)*x^k ).
The n-th row of the new triangle T(n,k) (0 <= k <= n) gives the coefficients in the expansion of p(n+2).
The new triangle may be defined recursively by: T(n,0)=1; T(n,k) = T(n,k-1) + (-1)^k*U(n+2,k) for 1 <= k <= floor(n/2); T(n,k) = T(n,n-k).
Note that the central terms in the odd-numbered rows of U(n,k) do not get used.
The following table lists various sequences constructed using this transform:
Parameter Triangle  Triangle  Odd-numbered
m             U         T        rows
0          A007312  A007312   A034870
1          A008292  A159041   A171692
2          A060187  A225356   A225076
3          A142458  A225433   A225398
4          A142459  A225434   A225415

			Triangle begins as follows:
  1;
  1,     1;
  1,   -10,      1;
  1,   -25,    -25,        1;
  1,   -56,    246,      -56,       1;
  1,  -119,   1072,     1072,    -119,       1;
  1,  -246,   4047,   -11572,    4047,    -246,        1;
  1,  -501,  14107,   -74127,  -74127,   14107,     -501,      1;
  1, -1012,  46828,  -408364,  901990, -408364,    46828,  -1012,     1;
  1, -2035, 150602, -2052886, 7685228, 7685228, -2052886, 150602, -2035, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Roger L. Bagula, Another Mathematica program for A159041.

Cf. A007312, A008292, A034870, A060187, A142458, A142459, A159041, A171692, A225076, A225356, A225398, A225415, A225433, A225434.

A008292 := proc(n, k) option remember; if k < 1 or k > n then 0; elif k = 1 or k = n then 1; else k*procname(n-1, k)+(n-k+1)*procname(n-1, k-1) ; end if; end proc:
# row n of new triangle T(n,k) in terms of old triangle U(n,k):
p:=proc(n) local k; global U;
simplify( (1/(1-x)) * ( add((-1)^k*U(n+2,k+1)*x^k,k=0..floor(n/2)) + add((-1)^(n+k)*U(n+2,k+1)*x^k, k=ceil((n+2)/2)..n+1 )) );
end;
U:=A008292;
for n from 0 to 6 do lprint(simplify(p(n))); od: # N. J. A. Sloane, May 11 2013
A159041 := proc(n, k)
    if k = 0 then
        1;
    elif k <= floor(n/2) then
        A159041(n, k-1)+(-1)^k*A008292(n+2, k+1) ;
    else
        A159041(n, n-k) ;
    end if;
end proc: # R. J. Mathar, May 08 2013

A[n_, 1] := 1;
A[n_, n_] := 1;
A[n_, k_] := (n - k + 1)A[n - 1, k - 1] + k A[n - 1, k];
p[x_, n_] = Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], (-1)^i*A[n, i], -(-1)^(n - i)*A[n, i]]], {i, 0, n}]/(1 - x);
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
Flatten[%]

def A008292(n,k): return sum( (-1)^j*(k-j)^n*binomial(n+1,j) for j in (0..k) )
@CachedFunction
def T(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return T(n,k-1) + (-1)^k*A008292(n+2,k+1)
    else: return T(n,n-k)
flatten([[T(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*A008292(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1. - R. J. Mathar, May 08 2013

Edited by N. J. A. Sloane, May 07 2013, May 11 2013

A225076 Triangle read by rows: absolute values of odd-numbered rows of A225356.

Original entry on oeis.org

1, 1, 22, 1, 1, 236, 1446, 236, 1, 1, 2178, 58479, 201244, 58479, 2178, 1, 1, 19672, 1736668, 19971304, 49441990, 19971304, 1736668, 19672, 1, 1, 177134, 46525293, 1356555432, 9480267666, 19107752148, 9480267666, 1356555432, 46525293, 177134, 1

Offset: 1

Roger L. Bagula, Apr 26 2013

An equivalent definition: take the polynomials corresponding to rows 2, 4, 6, 8, ... of A060187, divide by x+1, and extract the coefficients. [Corrected by Petros Hadjicostas, Apr 17 2020]

			Triangle T(n,m) (for n >= 1 and 0 <= m <= 2*n - 2) begins as follows:
  1;
  1,    22,       1;
  1,   236,    1446,      236,        1;
  1,  2178,   58479,   201244,    58479,     2178,       1;
  1, 19672, 1736668, 19971304, 49441990, 19971304, 1736668, 19672, 1;
  ...

G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened

Cf. A002671 (row sums), A034870, A060187, A171692, A225398.

(* Power series via an infinite sum *)
p[x_,n_] = (x-1)^(2*n)*Sum[(2*k+1)^(2*n-1)*x^k,{k,0,Infinity}];
Table[CoefficientList[p[x,n]/(1+x),x],{n,1,10}]//Flatten
(* First alternative method: recurrence *)
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]];
T[n_, k_]:= T[n, k]= t[n+1,k+1,2]; (* t(n,k,2) = A060187 *)
Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(x+1), x], {n,14,2}]]
(* Second alternative method: polynomial expansion *)
p[t_] = Exp[t]*x/(-Exp[2*t] + x);
Flatten[Table[CoefficientList[(n!*(-1+x)^(n+1)/(x*(x+1)))*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 1, 13, 2}]]

def A060187(n, k): return sum( (-1)^(k-j)*(2*j-1)^(n-1)*binomial(n, k-j) for j in (1..k) )
def A225076(n,k): return sum( (-1)^(k-j-1)*A060187(2*n, j+1) for j in (0..k-1) )
flatten([[A225076(n, k) for k in (1..2*n-1)] for n in (1..12)]) # G. C. Greubel, Mar 19 2022

Triangle read by rows: row n gives coefficients in the expansion of the polynomial ((x - 1)^(2*n)/(x + 1)) * Sum_{k >= 0} (2*k + 1)^(2*n-1)*x^k. The infinite sum simplifies to a polynomial.
Sum_{m=0..2*n-2} T(n,m)*t^m = 2^(2*n-1) * (1-t)^(2*n) * LerchPhi(t, 1-2*n, 1/2)/(1 + t).
Sum_{k=1..n} T(n, k) = A002671(n-1).
T(n,m) = Sum_{k=0..m-1} (-1)^(m-k-1)*A060187(2*n,k+1) for n >= 1 and 1 <= m <= 2*n-1. - Petros Hadjicostas, Apr 17 2020

Edited by N. J. A. Sloane, May 06 2013, May 11 2013

A225398 Triangle read by rows: absolute values of odd-numbered rows of A225433.

Original entry on oeis.org

1, 1, 38, 1, 1, 676, 4806, 676, 1, 1, 10914, 362895, 1346780, 362895, 10914, 1, 1, 174752, 20554588, 263879264, 683233990, 263879264, 20554588, 174752, 1, 1, 2796190, 1063096365, 35677598760, 267248150610, 554291429748, 267248150610, 35677598760, 1063096365, 2796190, 1

Offset: 1

Roger L. Bagula, Apr 26 2013 (Entered by N. J. A. Sloane, May 06 2013)

			Triangle begins:
  1;
  1,     38,        1;
  1,    676,     4806,       676,         1;
  1,  10914,   362895,   1346780,    362895,     10914,        1;
  1, 174752, 20554588, 263879264, 683233990, 263879264, 20554588, 174752, 1;

G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened

Cf. A034870, A171692, A225076.

(* First 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]];
T[n_, k_]:= T[n, k]= t[n+1, k+1, 3]; (* t(n,k,3) = A142458 *)
Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(1+x), x], {n, 1, 14, 2}]]
(* 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]]; (* t(n,k,3) = A142458 *)
A225398[n_, k_]:= A225398[n, k]= Sum[(-1)^(k-j-1)*t[2*n,j+1,3], {j,0,k-1}];
Table[A225398[n, k], {n,12}, {k,2*n-1}] //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)
def A225398(n,k): return sum( (-1)^(k-j-1)*A142458(2*n, j+1) for j in (0..k-1) )
flatten([[A225398(n, k) for k in (1..2*n-1)] for n in (1..12)]) # G. C. Greubel, Mar 19 2022

From G. C. Greubel, Mar 19 2022: (Start)
T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142458(2*n, j+1).
T(n, n-k) = T(n, k). (End)

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

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

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

A225415 Triangle read by rows: absolute values of odd-numbered rows of A225434.

Original entry on oeis.org

1, 1, 58, 1, 1, 1556, 12006, 1556, 1, 1, 39054, 1461615, 5647300, 1461615, 39054, 1, 1, 976552, 135028828, 1838120344, 4873361350, 1838120344, 135028828, 976552, 1, 1, 24414050, 11462721645, 414730580760, 3221733789330, 6783391017228, 3221733789330, 414730580760, 11462721645, 24414050, 1

Offset: 1

Roger L. Bagula, May 07 2013

			Triangle begins:
  1;
  1,     58,         1;
  1,   1556,     12006,       1556,          1;
  1,  39054,   1461615,    5647300,    1461615,      39054,         1;
  1, 976552, 135028828, 1838120344, 4873361350, 1838120344, 135028828, 976552, 1;

G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened

The m=4 triangle in the sequence A034870 (m=0), A171692 (m=1), A225076 (m=2), A225398 (m=3).

(* First 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]];
T[n_, k_]:= T[n, k] = t[n+1, k+1,4]; (* t(n,k,4) = A142459 *)
Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(1+x), x], {n,1,14,2}]]
(* 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]]; (* t(n,k,4) = A142459 *)
T[n_, k_]:= T[n, k]= Sum[ (-1)^(k-j-1)*t[2*n,j+1,4], {j,0,k-1}];
Table[T[n, k], {n,12}, {k,2*n-1}]//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)
def A225415(n,k): return sum( (-1)^(k-j-1)*A142459(2*n, j+1) for j in (0..k-1) )
flatten([[A225415(n, k) for k in (1..2*n-1)] for n in (1..12)]) # G. C. Greubel, Mar 19 2022

T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142459(2*n, j+1). - G. C. Greubel, Mar 19 2022

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

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

cp's OEIS Frontend

A159041 Triangle read by rows: row n (n>=0) gives the coefficients of the polynomial p(n,x) of degree n defined in comments.

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A225076 Triangle read by rows: absolute values of odd-numbered rows of A225356.

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A225398 Triangle read by rows: absolute values of odd-numbered rows of A225433.

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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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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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A225415 Triangle read by rows: absolute values of odd-numbered rows of A225434.

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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.

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.