A109128 -id:A109128 - cp's OEIS frontend

A131067 Triangle read by rows: T(n,k) = 7*binomial(n,k) - 6 for 0 <= k <= n.

1, 1, 1, 1, 8, 1, 1, 15, 15, 1, 1, 22, 36, 22, 1, 1, 29, 64, 64, 29, 1, 1, 36, 99, 134, 99, 36, 1, 1, 43, 141, 239, 239, 141, 43, 1, 1, 50, 190, 386, 484, 386, 190, 50, 1, 1, 57, 246, 582, 876, 876, 582, 246, 57, 1, 1, 64, 309, 834, 1464, 1758, 1464, 834, 309, 64, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A131068: (1, 2, 10, 32, 82, 188, 406, ...), the binomial transform of (1, 1, 7, 7, 7, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  8,  1;
  1, 15, 15,  1;
  1, 22, 36, 22,  1;
  1, 29, 64, 64, 29, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A131064, A131066.

Sequence m*binomial(n,k) - (m-1): A007318 (m=1), A109128 (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), this sequence (m=7), A131068 (m=8).

[7*Binomial(n, k) -6: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

T := proc (n, k) if k <= n then 7*binomial(n, k)-6 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007

Table[7*Binomial[n, k] -6, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[7*binomial(n, k) -6 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

G.f.: G(t,z) = (1-z-t*z+7*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007

More terms from Emeric Deutsch, Jun 20 2007

A123203 a(n) = 2^(n+1) - 3*n.

Original entry on oeis.org

1, 2, 7, 20, 49, 110, 235, 488, 997, 2018, 4063, 8156, 16345, 32726, 65491, 131024, 262093, 524234, 1048519, 2097092, 4194241, 8388542, 16777147, 33554360, 67108789, 134217650, 268435375, 536870828, 1073741737, 2147483558

Offset: 1

Gary W. Adamson, Jun 13 2007

An elephant sequence, see A175654. For the corner squares just one A[5] vector, with decimal value 186, leads to this sequence. For the central square this vector leads to the companion sequence A036563. - Johannes W. Meijer, Aug 15 2010

			a(4) = 20, row sums of 4th row of triangle A131062: (1, 9, 9, 1).
a(4) = 20 = (1, 3, 3, 1) dot (1, 1, 4, 4) = (1 + 3 + 12 + 4).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Joseph Breen and Emma Copeland, Non-orientable Nurikabe, arXiv:2506.12612 [math.CO], 2025. See pp. 1, 4.
Tamas Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015) 73-94.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000079, A000295, A109128, A131060, A131061, A131063, A131064, A131065, A131066.

[2^(n+1) -3*n: n in [1..40]]; // G. C. Greubel, Sep 14 2024

Table[2^(n+1) - 3*n, {n,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
LinearRecurrence[{4,-5,2},{1,2,7},40] (* Harvey P. Dale, Mar 30 2024 *)

def A123203(n): return 2^(n+1) -3*n
[A123203(n) for n in range(1,41)] # G. C. Greubel, Sep 14 2024

Binomial transform of [1, 1, 4, 4, 4, ...].
Equals row sums of triangle A131061.
From Johannes W. Meijer, Aug 15 2010; corrected by Colin Barker, Jul 28 2012: (Start)
a(n) = 2^(1+n) - 3*n.
a(n) = 3*A000295(n-1) + A000079(n-1).
(End)
G.f.: x*(1 - 2*x + 4*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Jul 28 2012
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Colin Barker, Jul 29 2012
E.g.f.: 2*exp(2*x) - 3*x*exp(x) - 2. - G. C. Greubel, Sep 14 2024

More terms from Vladimir Joseph Stephan Orlovsky, Nov 15 2008

Title changed by G. C. Greubel, Sep 14 2024

A131064 Binomial transform of [1, 1, 5, 5, 5, ...].

Original entry on oeis.org

1, 2, 8, 24, 60, 136, 292, 608, 1244, 2520, 5076, 10192, 20428, 40904, 81860, 163776, 327612, 655288, 1310644, 2621360, 5242796, 10485672, 20971428, 41942944, 83885980, 167772056, 335544212, 671088528, 1342177164, 2684354440

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums of triangle A131063. - Emeric Deutsch, Jun 20 2007

			a(3) = 24 = sum of row 4 terms of A131063: (1 + 11 + 11 + 1).
a(3) = 24 = (1, 3, 3, 1) dot (1, 1, 5, 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A109128, A123203, A131060, A131061, A131063, A131065, A131066, A131067, A131068.

Print(List([0..30],n->5*2^n-4*n-4)); # Muniru A Asiru, Feb 21 2019

I:=[1, 2, 8]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2) + 2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 05 2012

a := proc (n) options operator, arrow; 5*2^n-4*n-4 end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Jun 20 2007

CoefficientList[Series[(1-2x+5x^2)/((1-2x)(1-x)^2),{x,0,40}],x] (* Vincenzo Librandi, Jul 05 2012 *)
LinearRecurrence[{4,-5,2},{1,2,8},30] (* Harvey P. Dale, Dec 29 2014 *)

[5*2^n -4*(n+1) for n in (0..30)] # G. C. Greubel, Mar 12 2020

From Emeric Deutsch, Jun 20 2007: (Start)
a(n) = 5*2^n - 4*(n + 1).
G.f.: (1-2*x+5*x^2)/((1-2*x)*(1-x)^2). (End)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Vincenzo Librandi, Jul 05 2012
E.g.f.: 5*exp(2*x) - 4*(1+x)*exp(x). - G. C. Greubel, Mar 12 2020

Corrected and extended by Emeric Deutsch, Jun 20 2007

A131066 Binomial transform of [1, 1, 6, 6, 6, ...].

Original entry on oeis.org

1, 2, 9, 28, 71, 162, 349, 728, 1491, 3022, 6089, 12228, 24511, 49082, 98229, 196528, 393131, 786342, 1572769, 3145628, 6291351, 12582802, 25165709, 50331528, 100663171, 201326462, 402653049, 805306228, 1610612591, 3221225322

Offset: 0

Gary W. Adamson, Jun 13 2007

nonn

Row sums of triangle A131065. - Emeric Deutsch, Jun 20 2007

			a(3) = 28 = sum of row 4 of triangle A131065: (1 + 13 + 13 + 1).
a(3) = 28 = (1, 3, 3, 1) dot (1, 1, 6, 6) = (1 + 3 + 18 + 6).

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A109128, A123203, A131060, A131061, A131063, A131064, A131065, A131067, A131068.

Print(List([0..30],n->6*2^n-5*n-5)); # Muniru A Asiru, Feb 21 2019

[6*2^n -5*(n+1): n in [0..30]]; // G. C. Greubel, Mar 12 2020

a := proc (n) options operator, arrow; 6*2^n-5*n-5 end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Jun 20 2007

Table[6*2^n -5*(n+1), {n,0,30}] (* G. C. Greubel, Mar 12 2020 *)

[6*2^n -5*(n+1) for n in (0..30)] # G. C. Greubel, Mar 12 2020

From Emeric Deutsch, Jun 20 2007: (Start)
a(n) = 6*2^n - 5*(n + 1).
G.f.: (1 - 2*x + 6*x^2)/((1-2*x)*(1-x)^2). (End)
E.g.f.: 6*exp(2*x) - 5*(1 + x)*exp(x). - G. C. Greubel, Mar 12 2020
a(n) = 2*a(n - 1) + 5*n - 5. - Kritsada Moomuang, Jul 03 2020
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, Jul 10 2020

Corrected and extended by Emeric Deutsch, Jun 20 2007

A168625 Triangle T(n,k) = 8*binomial(n,k) - 7 with columns 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 17, 17, 1, 1, 25, 41, 25, 1, 1, 33, 73, 73, 33, 1, 1, 41, 113, 153, 113, 41, 1, 1, 49, 161, 273, 273, 161, 49, 1, 1, 57, 217, 441, 553, 441, 217, 57, 1, 1, 65, 281, 665, 1001, 1001, 665, 281, 65, 1, 1, 73, 353, 953, 1673, 2009, 1673, 953, 353, 73, 1

Offset: 0

Roger L. Bagula, Dec 01 2009

Triangle T(n,k): the coefficient [x^k] of the polynomial 8*(x+1)^n -7*( x^(n+1) - 1)/(x-1).

			Triangle begins as:
  1;
  1,  1;
  1,  9,   1;
  1, 17,  17,   1;
  1, 25,  41,  25,    1;
  1, 33,  73,  73,   33,    1;
  1, 41, 113, 153,  113,   41,    1;
  1, 49, 161, 273,  273,  161,   49,   1;
  1, 57, 217, 441,  553,  441,  217,  57,   1;
  1, 65, 281, 665, 1001, 1001,  665, 281,  65,  1;
  1, 73, 353, 953, 1673, 2009, 1673, 953, 353, 73, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Sequence m*binomial(n,k) - (m-1): A007318 (m=1), A109128 (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), A131067 (m=7), this sequence (m=8).

[8*Binomial(n, k) -7: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

A168625:= (n,k) -> 8*binomial(n, k) -7; seq(seq(A168625(n, k), k = 0..n), n = 0.. 10); # G. C. Greubel, Mar 12 2020

m = 8; p[x_, n_]:= FullSimplify[ExpandAll[m*(x+1)^n -(m-1)(x^(n+1) -1)/(x-1)]];
Table[CoefficientList[p[x, n], x], {n,0,10}]//Flatten
Table[8*Binomial[n, k] -7, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[8*binomial(n, k) -7 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

T(n,k) = [x^k] ( 8*(x+1)^n-7*Sum_{s=0..n} x^s ) = 8*A007318(n,k) - 7. - R. J. Mathar, Sep 02 2011

Definition simplified by R. J. Mathar, Sep 02 2011

A176200 A symmetrical triangle T(n, m) = 2*Eulerian(n+1, m) -1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 21, 21, 1, 1, 51, 131, 51, 1, 1, 113, 603, 603, 113, 1, 1, 239, 2381, 4831, 2381, 239, 1, 1, 493, 8585, 31237, 31237, 8585, 493, 1, 1, 1003, 29215, 176467, 312379, 176467, 29215, 1003, 1, 1, 2025, 95679, 910383, 2620707, 2620707, 910383, 95679, 2025, 1

Offset: 0

Roger L. Bagula, Apr 11 2010

Row sums are: {1, 2, 9, 44, 235, 1434, 10073, 80632, 725751, 7257590, 79833589, ...}.

			Triangle begins as:
  1;
  1,   1;
  1,   7,    1;
  1,  21,   21,     1;
  1,  51,  131,    51,     1;
  1, 113,  603,   603,   113,    1;
  1, 239, 2381,  4831,  2381,  239,   1;
  1, 493, 8585, 31237, 31237, 8585, 493, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A008292, A109128.

Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
[[2*Eulerian(n+1,k)-1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Apr 25 2019

Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j,0,k+1}];
T[n_, m_]:= 2*Eulerian[n+1, m]-1;
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten (* modified by G. C. Greubel, Apr 25 2019 *)

Eulerian(n,k) = sum(j=0,k+1, (-1)^j*binomial(n+1,j)*(k-j+1)^n); {T(n,k) = 2*Eulerian(n+1,k) - 1 };
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Apr 25 2019

def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
def T(n,k): return 2*Eulerian(n+1,k)-1
[[T(n,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Apr 25 2019

T(n, m) = 2*Eulerian(n+1, m) - 1, where Eulerian(n, k) = A008292(n,k).

Edited by G. C. Greubel, Apr 25 2019

A132737 Triangle T(n,k) = 2*binomial(n,k) + 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 7, 7, 1, 1, 9, 13, 9, 1, 1, 11, 21, 21, 11, 1, 1, 13, 31, 41, 31, 13, 1, 1, 15, 43, 71, 71, 43, 15, 1, 1, 17, 57, 113, 141, 113, 57, 17, 1, 1, 19, 73, 169, 253, 253, 169, 73, 19, 1, 1, 21, 91, 241, 421, 505, 421, 241, 91, 21, 1, 1, 23, 111, 331, 661, 925, 925, 661, 331, 111, 23, 1

Offset: 0

Gary W. Adamson, Aug 26 2007

			First few rows of the triangle are:
  1;
  1,  1;
  1,  5,  1;
  1,  7,  7,  1;
  1,  9, 13,  9,  1;
  1, 11, 21, 21, 11,  1;
  1, 13, 31, 41, 31, 13,  1;
  1, 15, 43, 71, 71, 43, 15, 1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A132735, A132738.

Sequences of the form 2*binomial(n,k) + q: A132729 (q=-3), A132731 (q=-2), A109128 (q=-1), A132046 (q=0), this sequence (q=1).

A132737:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) +1 >;
[A132737(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 15 2021

T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n,k] +1];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)

def A132737(n,k): return 1 if (k==0 or k==n) else 2*binomial(n,k) + 1
flatten([[A132737(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 15 2021

T(n, k) = 2*A132735(n, k) - 1, an infinite lower triangular matrix.
T(n,0) = T(n,n) = 1; otherwise T(n,k) = 2*C(n,k) + 1. - Franklin T. Adams-Watters, Jul 06 2009
Sum_{k=0..n} T(n, k) = 2^(n+1) + n - 3 + 2*[n=0] = A132738(n). - G. C. Greubel, Feb 15 2021

Extended by Franklin T. Adams-Watters, Jul 06 2009

A132752 Triangle T(n, k) = 2*A132749(n, k) - 1, read by rows.

Original entry on oeis.org

1, 3, 1, 3, 3, 1, 3, 5, 5, 1, 3, 7, 11, 7, 1, 3, 9, 19, 19, 9, 1, 3, 11, 29, 39, 29, 11, 1, 3, 13, 41, 69, 69, 41, 13, 1, 3, 15, 55, 111, 139, 111, 55, 15, 1, 3, 17, 71, 167, 251, 251, 167, 71, 17, 1

Offset: 0

Gary W. Adamson, Aug 28 2007

			First few rows of the triangle are:
  1;
  3,  1;
  3,  3,  1;
  3,  5,  5,  1;
  3,  7, 11,  7,  1;
  3,  9, 19, 19,  9,  1;
  3, 11, 29, 39, 29, 11, 1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A109128, A132749, A132753.

A132752:= func< n,k | k eq n select 1 else k eq 0 select 3 else 2*Binomial(n,k) -1 >;
[A132752(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

T[n_, k_]:= If[k==n, 1, If[k==0, 3, 2*Binomial[n, k] -1 ]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

def A132752(n,k): return 1 if k==n else 3 if k==0 else 2*binomial(n,k) -1
flatten([[A132752(n,k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Feb 16 2021

T(n, k) = 2*A132749(n, k) - 1, an infinite lower triangular matrix.
From G. C. Greubel, Feb 16 2021: (Start)
T(n, k) = A109128(n, k) with T(n, 0) = 3.
Sum_{k=0..n} T(n, k) = 2^(n+1) -n +1 -2*[n=0] = A132753(n) - 2*[n=0]. (End)

A141596 Triangle T(n,k) = 4*binomial(n,k)^2 - 3, read by rows, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 13, 1, 1, 33, 33, 1, 1, 61, 141, 61, 1, 1, 97, 397, 397, 97, 1, 1, 141, 897, 1597, 897, 141, 1, 1, 193, 1761, 4897, 4897, 1761, 193, 1, 1, 253, 3133, 12541, 19597, 12541, 3133, 253, 1, 1, 321, 5181, 28221, 63501, 63501, 28221, 5181, 321, 1, 1, 397, 8097, 57597, 176397, 254013, 176397, 57597, 8097, 397, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Aug 21 2008

			Triangle begins as:
  1;
  1,   1;
  1,  13,    1;
  1,  33,   33,     1;
  1,  61,  141,    61,      1;
  1,  97,  397,   397,     97,      1;
  1, 141,  897,  1597,    897,    141,      1;
  1, 193, 1761,  4897,   4897,   1761,    193,     1;
  1, 253, 3133, 12541,  19597,  12541,   3133,   253,    1;
  1, 321, 5181, 28221,  63501,  63501,  28221,  5181,  321,   1;
  1, 397, 8097, 57597, 176397, 254013, 176397, 57597, 8097, 397,  1;

Harvey P. Dale, Table of n, a(n) for n = 0..10000

Cf. A109128.

A141596:= func< n,k | 4*Binomial(n,k)^2 - 3 >;
[A141596(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 15 2024

Table[4*Binomial[n,k]^2-3,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Dec 21 2016 *)

def A141596(n,k): return 4*binomial(n,k)^2 -3
flatten([[A141596(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 15 2024

Sum_{k=0..n} T(n, k) = 4*binomial(2*n,n) - 3*(n+1) (row sums).

Sum_{k=0..n} (-1)^k*T(n, k) = ((1 + (-1)^n)/2)*(4*(-1)^(n/2)*binomial(n, n/2) - 3) (alternating sign row sums). - G. C. Greubel, Sep 15 2024

A141591 Triangle, read by rows, T(n, k) = 2*A123125(n-1, k), for n >= 2, otherwise T(n, 0) = T(n, n) = -1, with T(0, 0) = T(1, 0) = 1.

Original entry on oeis.org

1, 1, -1, -1, 2, -1, -1, 2, 2, -1, -1, 2, 8, 2, -1, -1, 2, 22, 22, 2, -1, -1, 2, 52, 132, 52, 2, -1, -1, 2, 114, 604, 604, 114, 2, -1, -1, 2, 240, 2382, 4832, 2382, 240, 2, -1, -1, 2, 494, 8586, 31238, 31238, 8586, 494, 2, -1, -1, 2, 1004, 29216, 176468, 312380, 176468, 29216, 1004, 2, -1, -1, 2, 2026, 95680, 910384, 2620708, 2620708, 910384, 95680, 2026, 2, -1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Aug 20 2008

			Triangle begins as:
   1;
   1, -1;
  -1,  2,   -1;
  -1,  2,    2,    -1;
  -1,  2,    8,     2,     -1;
  -1,  2,   22,    22,      2,     -1;
  -1,  2,   52,   132,     52,      2,     -1;
  -1,  2,  114,   604,    604,    114,      2,    -1;
  -1,  2,  240,  2382,   4832,   2382,    240,     2,   -1;
  -1,  2,  494,  8586,  31238,  31238,   8586,   494,    2,  -1;
  -1,  2, 1004, 29216, 176468, 312380, 176468, 29216, 1004,   2,  -1;

Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, McGraw-Hill, New York, 1976, page 91.

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

Cf. 033312, A109128.

Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j)^n: j in [0..k]]) >; // A008292
function A141591(n,k)
  if n eq 0 then return 1;
  elif k eq 0 and n eq 1 then return 1;
  elif k eq 0 or k eq n then return -1;
  else return 2*Eulerian(n-1,k);
  end if;
end function;
[A141591(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 15 2024

(* First program *)
f[x_, n_]:= f[x, n]= (1-x)^(n+1)*Sum[k^n*x^k, {k, 0, Infinity}];
Table[Simplify[f[x, n]], {n, 0, 10}];
Join[{{1}}, Table[Join[CoefficientList[2*f[x,n] -1, x], {-1}], {n, 0, 10}]]//Flatten
(* Second program *)
Eulerian[n_, k_]:= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}]; (* A008292 *)
A141591[n_, k_]:= If[k==0 || k==n, -1, 2*Eulerian[n-1,k]] +2*Boole[n==0 || n ==1 && k==0];
Table[A141591[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 15 2024 *)

@CachedFunction
def A008292(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j)^n for j in range(k+1))
def A141591(n,k):
    if (k==0 and n==0): return 1
    elif (k==0 and n==1): return 1
    elif (k==0 or k==n): return -1
    else: return 2*A008292(n-1, k)
flatten([[A141591(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 15 2024

T(n, k) = 2*A123125(n-1, k), with T(0, 0) = T(1, 0) = 1, otherwise T(n, 0) = T(n, n) = -1.
Sum_{k=0..n} T(n, k) = 2*033312(n), for n >= 1, otherwise 1 (n=0).
From G. C. Greubel, Sep 15 2024: (Start)
T(n, k) = 2*A008292(n, k) for n >= 2, 1 <= k <= n-1, with T(n, 0) = T(n, n) = -1, T(0, 0) = T(1, 0) = 1.
T(n, n-k) = T(n, k) for n >= 2. (End)

Edited and new name by G. C. Greubel, Sep 15 2024

cp's OEIS Frontend

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