A142459 -id:A142459 - cp's OEIS frontend

A155493 Triangle T(n, k) = binomial(n+1, k)*A142459(n+1, k+1)/(k+1), read by rows.

1, 1, 1, 1, 15, 1, 1, 118, 118, 1, 1, 770, 3540, 770, 1, 1, 4671, 67810, 67810, 4671, 1, 1, 27321, 1039689, 3085355, 1039689, 27321, 1, 1, 156220, 14006244, 99524810, 99524810, 14006244, 156220, 1, 1, 878868, 173788752, 2602528824, 6090918372, 2602528824, 173788752, 878868, 1

Offset: 0

Roger L. Bagula, Jan 23 2009

			Triangle begins as:
  1;
  1,      1;
  1,     15,         1;
  1,    118,       118,          1;
  1,    770,      3540,        770,          1;
  1,   4671,     67810,      67810,       4671,          1;
  1,  27321,   1039689,    3085355,    1039689,      27321,         1;
  1, 156220,  14006244,   99524810,   99524810,   14006244,    156220,      1;
  1, 878868, 173788752, 2602528824, 6090918372, 2602528824, 173788752, 878868, 1;

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

Cf. A001263 (m=0), A155467 (m=1), A155491 (m=3), this sequence (m=4).

Cf. A142459.

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_, m_]:= Binomial[n+1,k]*t[n+1,k+1,m]/(k+1);
Table[T[n,k,4], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 01 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 T(n,k,m): return binomial(n+1,k)*t(n+1,k+1,m)/(k+1)
flatten([[T(n,k,4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 01 2022

T(n, k) = binomial(n+1, k)*t(n, k, m)/(k+1), where t(n,k,m) = (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m), t(n,1,m) = t(n,n,m) = 1, and m = 4.
From G. C. Greubel, Apr 01 2022: (Start)
T(n, k) = binomial(n+1, k)*A142459(n+1, k+1)/(k+1).
T(n, n-k) = T(n, k). (End)

Edited by G. C. Greubel, Apr 01 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

A172013 a(n) = 6A142459(2n, n)/(n+1).

Original entry on oeis.org

3, 118, 20343, 8530698, 6711481694, 8575821262764, 16243345162977759, 42826533033277249154, 150138953276380791799098, 675925071086215282939520628, 3802445616812067139270851537718, 26147695687370407271086390933321188, 215852465255521412471161891166554453788

Offset: 1

Roger L. Bagula, Nov 19 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 1..180

Cf. A142458, A142459, A172010.

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]];
A142459[n_, k_]:= A142459[n, k]= T[n,k,4];
A172013[n_]:= A172013[n]= 6*A142459[2*n, n]/(n+1);
Table[A172013[n], {n,30}] (* modified by G. C. Greubel, Mar 18 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 A172013(n): return 6*A142459(2*n, n)/(n+1)
[A172013(n) for n in (1..30)] # G. C. Greubel, Mar 18 2022

a(n) = 6*A142459(2*n, n)/(n+1).

Offset and formula corrected by G. C. Greubel, Mar 18 2022

A167787 Triangle of z Transform coefficients from General Pascal [1,10,1} A142459 polynomials multiplied by factor 3^Floor[(2*k - 1)/3].

Original entry on oeis.org

0, 3, 3, 6, 9, 54, 54, 27, 324, 810, 540, 27, 432, 2322, 3780, 1890, 81, 810, 12150, 42120, 51030, 20412, 243, 3402, 27216, 272160, 697410, 673596, 224532, 243, 34020, 40824, 244944, 1786050, 3633336, 2918916, 833976, 729, 104976, 1583388, 1224720

Offset: 0

Roger L. Bagula, Nov 12 2009

Row sums are:

{0, 3, 9, 117, 1701, 8451, 126603, 1898559, 9492309, 142383177, 2135743281...}

			{0},
{3},
{3, 6},
{9, 54, 54},
{27, 324, 810, 540},
{27, 432, 2322, 3780, 1890},
{81, 810, 12150, 42120, 51030, 20412},
{243, 3402, 27216, 272160, 697410, 673596, 224532},
{243, 34020, 40824, 244944, 1786050, 3633336, 2918916, 833976},
{729, 104976, 1583388, 1224720, 5664330, 32332608, 54561276, 37528920, 9382230},
{2187, -5734314, 6009876, 53905176, 31689630, 117756828, 551675124, 795613104, 478493730, 106331940}

Cf. A142458, A060187, A008292, A142459

m = 4; A[n_, 1] := 1; A[n_, n_] := 1
A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]
a = Table[A[n, k], {n, 10}, {k, n}]
p[x_, n_] = x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)
b = Table[p[x, n], {n, 0, 10}]
Table[3^Floor[(2*k - 1)/3]*CoefficientList[ExpandAll[ InverseZTransform[b[[k]], x, n] /. UnitStep[ -1 + n] -> 1], n], {k, 1, Length[b]}]

m=4;
A(n,k)= (m*n - m*k + 1)A(n - 1, k - 1} + (m*k - (m - 1))A(n - 1, k)
q(n,k)=InverseZTransform[x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)^n, x, k]
out_n,k=3^Floor[(2*k - 1)/3]*coefficients(q[n,k])

A172014 A142459(2*n,n)/(n+1).

Original entry on oeis.org

1, 5, 354, 88153, 48340622, 46980371858, 71465177189700, 157019003242118337, 471091863366049740694, 1851713757075363098855542

Offset: 0

Roger L. Bagula, Nov 19 2010

nonn

Cf. A142459.

A[n_, 1] := 1; A[n_, n_] := 1;
A[n_, k_] := (4*n - 4*k + 1)A[n - 1, k - 1] + (4*k - 3)A[n - 1, k];
a = Table[A[n, k], {n, 20}, {k, n}];
Table[a[[2*n + 1, n + 1]]/(n + 1), {n, 0, 9}]

a(n) = A142459(2*n,n)/(n+1).

A142458 Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3n-3k+1)T(n-1,k-1) + (3k-2)*T(n-1,k).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 39, 39, 1, 1, 166, 546, 166, 1, 1, 677, 5482, 5482, 677, 1, 1, 2724, 47175, 109640, 47175, 2724, 1, 1, 10915, 373809, 1709675, 1709675, 373809, 10915, 1, 1, 43682, 2824048, 23077694, 44451550, 23077694, 2824048, 43682, 1

Offset: 1

Roger L. Bagula, Sep 19 2008

Consider the triangle T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k). For m = ...,-2,-1,0,1,2,3,... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, ... - N. J. A. Sloane, May 08 2013

			The rows n >= 1 and columns 1 <= k <= n look as follows:
  1;
  1,     1;
  1,     8,       1;
  1,    39,      39,        1;
  1,   166,     546,      166,        1;
  1,   677,    5482,     5482,      677,        1;
  1,  2724,   47175,   109640,    47175,     2724,       1;
  1, 10915,  373809,  1709675,  1709675,   373809,   10915,     1;
  1, 43682, 2824048, 23077694, 44451550, 23077694, 2824048, 43682, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_3(n,k).

Cf. A225372 (m=-2), A144431 (m=-1), A007318 (m=0), A008292 (m=1), A060187 (m=2), this sequence (m=3), A142459 (m=4), A142560 (m=5), A142561 (m=6), A142562 (m=7), A167884 (m=8), A257608 (m=9).

Cf. A008544, A142976, A144380, A144381, A144414.

A142458 := proc(n,k) if n = k then 1; elif k > n or k < 1 then 0 ;else (3*n-3*k+1)*procname(n-1,k-1)+(3*k-2)*procname(n-1,k) ; end if; end proc:
seq(seq(A142458(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Jun 04 2011

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] ];
Table[T[n, k, 3], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by G. C. Greubel, Mar 14 2022 *)

def T(n,k,m): # A142458
    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)
flatten([[T(n,k,3) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Mar 14 2022

T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = 3.
Sum_{k=1..n} T(n, k) = A008544(n-1).
From G. C. Greubel, Mar 14 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = A144414(n-1).
T(n, 3) = A142976(n-2).
T(n, 4) = A144380(n-3).
T(n, 5) = A144381(n-4). (End)

Edited by the Associate Editors of the OEIS, Aug 28 2009

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

Original entry on oeis.org

1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512

Offset: 0

Dale Gerdemann, Apr 12 2015

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:
  a\b 1.......2.......3.......4.......5.......6
  -1  A144431
  0   A007318 A038208 A038221
  1   A008292 A256890 A257180 A257606 A257607
  2   A060187 A257609 A257611 A257613 A257615
  3   A142458 A257610 A257620 A257622 A257624 A257626
  4   A142459 A257612 A257621
  5   A142460 A257614 A257623
  6   A142461 A257616 A257625
  7   A142462 A257617 A257627
  8   A167884 A257618
  9   A257608 A257619
The row sums of these, and similarly constructed number triangles, are shown in the following table:
  a\b 1.......2.......3.......4.......5.......6.......7.......8.......9
  0   A000079 A000302 A000400
  1   A000142 A001715 A001725 A049388 A049198
  2   A000165 A002866 A002866 A051580 A051582
  3   A008544 A051578 A037559 A051605 A051607 A051609
  4   A001813 A047053 A000407 A034177 A051618 A051620 A051622
  5   A047055 A008546 A008548 A034300 A034325 A051688 A051690
  6   A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103
  7   A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105
  8   A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115
  9   A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023
  10                                  A051262 A035265 A035273         A035277
  11                                  A254322
  12                                          A145448
The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).
  a\b  0           1           2          3
  -2   A130595/1
  -1
  0
  1    A110555/-1  A120434/-1  A144697/1  A144699/2
With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).
If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).
In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019

			Array, t(n, k), begins as:
   1,    2,      4,        8,        16,         32,          64, ...;
   2,   12,     52,      196,       684,       2276,        7340, ...;
   4,   52,    416,     2644,     14680,      74652,      357328, ...;
   8,  196,   2644,    26440,    220280,    1623964,    10978444, ...;
  16,  684,  14680,   220280,   2643360,   27227908,   251195000, ...;
  32, 2276,  74652,  1623964,  27227908,  381190712,  4677894984, ...;
  64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
    1;
    2,     2;
    4,    12,      4;
    8,    52,     52,       8;
   16,   196,    416,     196,      16;
   32,   684,   2644,    2644,     684,      32;
   64,  2276,  14680,   26440,   14680,    2276,     64;
  128,  7340,  74652,  220280,  220280,   74652,   7340,   128;
  256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172,   256;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. für die reine und angewandte Mathematik, 265 (1974): 110-37. See Section 3.
Dale Gerdemann, A256890, Plot of t(m,n) mod k , YouTube, 2015.
Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018-2019.

Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609.

Cf. A257610, A257612, A257614, A257616, A257617, A257618, A257619.

A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >;
[A256890(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022

Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j,0,k}], {n,0, 9}, {k,0,n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)

t(n,m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ Michel Marcus, Apr 14 2015

def A256890(n,k): return sum((-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n for j in range(k+1))
flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0 else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
Sum_{k=0..n} T(n, k) = A001715(n).
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(j+3,j)*binomial(n+4,k-j)*(j+2)^n. - Peter Bala, Dec 27 2019
Modified rule of Pascal: T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n else T(n,k) = f(n-k) * T(n-1,k-1) + f(k) * T(n-1,k), where f(x) = x + 2. - Georg Fischer, Nov 11 2021
From G. C. Greubel, Oct 18 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n). (End)

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

Original entry on oeis.org

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

A257612 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 4*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 24, 4, 8, 184, 184, 8, 16, 1216, 3680, 1216, 16, 32, 7584, 53824, 53824, 7584, 32, 64, 46208, 674752, 1507072, 674752, 46208, 64, 128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128, 256, 1677312, 84892672, 636233728, 1196803584, 636233728, 84892672, 1677312, 256

Offset: 0

Dale Gerdemann, May 06 2015

Corresponding entries in this triangle and in A060187 differ only by powers of 2. - F. Chapoton, Nov 04 2020

			Triangle begins as:
    1;
    2,      2;
    4,     24,       4;
    8,    184,     184,        8;
   16,   1216,    3680,     1216,       16;
   32,   7584,   53824,    53824,     7584,      32;
   64,  46208,  674752,  1507072,   674752,   46208,     64;
  128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128;

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

Cf. A047053 (row sums), A060187, A142459, A257621.
Cf. A038208, A256890, A257609, A257610, A257614, A257616, A257617, A257618, A257619.
See similar sequences listed in A256890.

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,4,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

f(x) = 4*x + 2;
T(n, k) = t(n-k, k);
t(n, m) = if (!n && !m, 1, if (n < 0 || m < 0, 0, f(m)*t(n-1,m) + f(n)*t(n,m-1)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print();); \\ Michel Marcus, May 06 2015

def T(n,k,a,b): # A257612
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,4,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.
Sum_{k=0..n} T(n,k) = A047053(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 4, and b = 2. - G. C. Greubel, Mar 20 2022

A142460 Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = 5.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 83, 83, 1, 1, 514, 1826, 514, 1, 1, 3105, 28310, 28310, 3105, 1, 1, 18656, 376615, 905920, 376615, 18656, 1, 1, 111967, 4627821, 22403635, 22403635, 4627821, 111967, 1, 1, 671838, 54377008, 478781506, 940952670, 478781506, 54377008, 671838, 1

Offset: 1

Roger L. Bagula, Sep 19 2008

One of a family of triangles. For m = ...,-2,-1,0,1,2,3,4,5,... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142560, ...

			Triangle begins as:
  1;
  1,      1;
  1,     12,        1;
  1,     83,       83,         1;
  1,    514,     1826,       514,         1;
  1,   3105,    28310,     28310,      3105,         1;
  1,  18656,   376615,    905920,    376615,     18656,        1;
  1, 111967,  4627821,  22403635,  22403635,   4627821,   111967,      1;
  1, 671838, 54377008, 478781506, 940952670, 478781506, 54377008, 671838, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_5(n,k).

Cf. A225372 (m=-2), A144431 (m=-1), A007318 (m=0), A008292 (m=1), A060187 (m=2), A142458 (m=3), A142459 (m=4), this sequence (m=5), A142561 (m=6), A142562 (m=7), A167884 (m=8), A257608 (m=9).

Cf. A047055 (row sums).

A142460 := proc(n, k) if n = k then 1; elif k > n or k < 1 then 0 ; else (5*n-5*k+1)*procname(n-1, k-1)+(5*k-4)*procname(n-1, k) ; end if; end proc:
seq(seq(A142459(n, k), k=1..n), n=1..10) ; # R. J. Mathar, May 11 2013

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] ];
Table[T[n, k, 5], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by G. C. Greubel, Mar 14 2022 *)

def T(n,k,m): # A142460
    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)
flatten([[T(n,k,5) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Mar 14 2022

T(n, k, m) = (m*n - m*k + 1)*T(n-1, k-1, m) + (m*k - (m-1))*T(n-1, k, m), with T(t,1,m) = T(n,n,m) = 1, and m = 5.

Sum_{k=1..n} T(n, k, 5) = A047055(n-1).

Edited by N. J. A. Sloane, May 08 2013, May 11 2013

cp's OEIS Frontend

A155493 Triangle T(n, k) = binomial(n+1, k)*A142459(n+1, k+1)/(k+1), read by rows.

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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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A172013 a(n) = 6*A142459(2*n, n)/(n+1).

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A167787 Triangle of z Transform coefficients from General Pascal [1,10,1} A142459 polynomials multiplied by factor 3^Floor[(2*k - 1)/3].

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A172014 A142459(2*n,n)/(n+1).

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A142458 Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

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A172013 a(n) = 6A142459(2n, n)/(n+1).

A142458 Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3n-3k+1)T(n-1,k-1) + (3k-2)*T(n-1,k).

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

A257612 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 4*x + 2.

A142460 Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = 5.