A038221 -id:A038221 - cp's OEIS frontend

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.

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)

A257620 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 3*n + 3.

Original entry on oeis.org

1, 3, 3, 9, 36, 9, 27, 297, 297, 27, 81, 2106, 5346, 2106, 81, 243, 13851, 73386, 73386, 13851, 243, 729, 87480, 868239, 1761264, 868239, 87480, 729, 2187, 540189, 9388791, 34158753, 34158753, 9388791, 540189, 2187, 6561, 3293622, 95843088, 578903274, 1024762590, 578903274, 95843088, 3293622, 6561

Offset: 0

Dale Gerdemann, May 09 2015

			Array t(n,k) begins as:
    1,      3,        9,         27,           81,            243, ...;
    3,     36,      297,       2106,        13851,          87480, ...;
    9,    297,     5346,      73386,       868239,        9388791, ...;
   27,   2106,    73386,    1761264,     34158753,      578903274, ...;
   81,  13851,   868239,   34158753,   1024762590,    25791697782, ...;
  243,  87480,  9388791,  578903274,  25791697782,   928501120152, ...;
  729, 540189, 95843088, 8959544136, 575025893586, 28788563928042, ...;
Triangle T(n,k) begins as:
     1;
     3,      3;
     9,     36,       9;
    27,    297,     297,       27;
    81,   2106,    5346,     2106,       81;
   243,  13851,   73386,    73386,    13851,     243;
   729,  87480,  868239,  1761264,   868239,   87480,    729;
  2187, 540189, 9388791, 34158753, 34158753, 9388791, 540189, 2187;

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

Cf. A000244, A008292, A034001 (row sums), A038221, A142458, A257180, A257610.
Cf. A257611, A257621, A257622, A257623, A257624, A257625, A257626, A257627.
Similar sequences listed in A256890.

A257620:= func< n,k | 3^n*EulerianNumber(n+1, k) >;
[A257620(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 17 2025

t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
T[n_, k_, p_, q_]= t[n-k, k, p, q];
Table[T[n,k,3,3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 28 2022 *)

from sage.all import *
from sage.combinat.combinat import eulerian_number
def A257620(n,k): return pow(3,n)*eulerian_number(n+1,k)
print(flatten([[A257620(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 17 2025

@CachedFunction
def t(n,k,p,q):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
def A257620(n,k): return t(n-k,k,3,3)
flatten([[A257620(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2022

T(n, k) = t(n-k, k), where 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), and f(n) = 3*n + 3.
Sum_{k=0..n} T(n, k) = A034001(n).
From G. C. Greubel, Feb 28 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)
T(n, k) = 3^n*A008292(n, k). - G. C. Greubel, Jan 17 2025

A257611 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(n) = 2*n + 3.

Original entry on oeis.org

1, 3, 3, 9, 30, 9, 27, 213, 213, 27, 81, 1308, 2982, 1308, 81, 243, 7431, 32646, 32646, 7431, 243, 729, 40314, 310263, 587628, 310263, 40314, 729, 2187, 212505, 2695923, 8701545, 8701545, 2695923, 212505, 2187, 6561, 1099704, 22059036, 113360904, 191433990, 113360904, 22059036, 1099704, 6561

Offset: 0

Dale Gerdemann, May 06 2015

			Array t(n,k) begins as:
    1,      3,        9,         27,          81,           243, ...;
    3,     30,      213,       1308,        7431,         40314, ...;
    9,    213,     2982,      32646,      310263,       2695923, ...;
   27,   1308,    32646,     587628,     8701545,     113360904, ...;
   81,   7431,   310263,    8701545,   191433990,    3579465642, ...;
  243,  40314,  2695923,  113360904,  3579465642,   93066106692, ...;
  729, 212505, 22059036, 1351133676, 59641127202, 2104476295026, ...;
Triangle T(n,k) begins as:
     1;
     3,      3;
     9,     30,       9;
    27,    213,     213,      27;
    81,   1308,    2982,    1308,      81;
   243,   7431,   32646,   32646,    7431,     243;
   729,  40314,  310263,  587628,  310263,   40314,    729;
  2187, 212505, 2695923, 8701545, 8701545, 2695923, 212505, 2187;

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

Cf. A000244, A051578 (row sums), A060187, A257609, A257613, A257615.
Cf. A038221, A257180, A257611, A257620, A257621, A257623, A257625, A257627.
Similar sequences listed in A256890.

t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
T[n_, k_, p_, q_]= t[n-k, k, p, q];
Table[T[n,k,2,3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 28 2022 *)

f(x) = 2*x + 3;
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

@CachedFunction
def t(n,k,p,q):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
def A257611(n,k): return t(n-k,k,2,3)
flatten([[A257611(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2022

T(n, k) = t(n-k, k), where 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), and f(n) = 2*n + 3.
Sum_{k=0..n} T(n, k) = A051578(n).
From G. C. Greubel, Feb 28 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)

A257180 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), and f(x) = x + 3.

Original entry on oeis.org

1, 3, 3, 9, 24, 9, 27, 141, 141, 27, 81, 726, 1410, 726, 81, 243, 3471, 11406, 11406, 3471, 243, 729, 15828, 81327, 136872, 81327, 15828, 729, 2187, 69873, 533259, 1390521, 1390521, 533259, 69873, 2187, 6561, 301362, 3295152, 12609198, 19467294, 12609198, 3295152, 301362, 6561, 19683, 1277619, 19489380, 105311556, 237144642, 237144642, 105311556, 19489380, 1277619, 19683

Offset: 0

Dale Gerdemann, Apr 17 2015

			Array t(n,k) begins as:
    1,     3,       9,        27,         81,         243, ... A000244;
    3,    24,     141,       726,       3471,       15828, ...;
    9,   141,    1410,     11406,      81327,      533259, ...;
   27,   726,   11406,    136872,    1390521,    12609198, ...;
   81,  3471,   81327,   1390521,   19467294,   237144642, ...;
  243, 15828,  533259,  12609198,  237144642,  3794314272, ...;
  729, 69873, 3295152, 105311556, 2607816498, 53824862658, ...;
Triangle T(n,k) begins as:
     1;
     3,      3;
     9,     24,       9;
    27,    141,     141,       27;
    81,    726,    1410,      726,       81;
   243,   3471,   11406,    11406,     3471,      243;
   729,  15828,   81327,   136872,    81327,    15828,     729;
  2187,  69873,  533259,  1390521,  1390521,   533259,   69873,   2187;
  6561, 301362, 3295152, 12609198, 19467294, 12609198, 3295152, 301362, 6561;

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

Cf. A000244, A008292, A001725 (row sums), A038221, A256890, A257606.
Cf. A257607, A257611, A257620, A257621, A257623, A257625, A257627.
Similar sequences listed in A256890.

f[n_]:= n+3;
t[n_, k_]:= t[n,k]= If[n<0 || k<0, 0, If[n==0 && k==0, 1, f[k]*t[n-1,k] +f[n]*t[n,k-1]]];
T[n_, k_]= t[n-k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 22 2022 *)

f(x) = x + 3;
T(n, k) = t(n-k, k);
t(n, m) = {if (!n && !m, return(1)); if (n < 0 || m < 0, return (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, Apr 23 2015

def f(n): return n+3
@CachedFunction
def t(n,k):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return f(k)*t(n-1, k) + f(n)*t(n, k-1)
def A257627(n,k): return t(n-k,k)
flatten([[A257627(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 22 2022

T(n,k) = t(n-k, k), where 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), and f(x) = x + 3.
Sum_{k=0..n} T(n, k) = A001725(n+5).
From G. C. Greubel, Feb 22 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)

A257621 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 4*n + 3.

Original entry on oeis.org

1, 3, 3, 9, 42, 9, 27, 393, 393, 27, 81, 3156, 8646, 3156, 81, 243, 23631, 142446, 142446, 23631, 243, 729, 171006, 2015895, 4273380, 2015895, 171006, 729, 2187, 1216725, 26107983, 102402705, 102402705, 26107983, 1216725, 2187, 6561, 8584872, 320039388, 2136524184, 3891302790, 2136524184, 320039388, 8584872, 6561

Offset: 0

Dale Gerdemann, May 09 2015

			Array t(n,k) begins as:
    1,       3,         9,          27,            81, ...;
    3,      42,       393,        3156,         23631, ...;
    9,     393,      8646,      142446,       2015895, ...;
   27,    3156,    142446,     4273380,     102402705, ...;
   81,   23631,   2015895,   102402705,    3891302790, ...;
  243,  171006,  26107983,  2136524184,  123074809242, ...;
  729, 1216725, 320039388, 40688926236, 3437022383970, ...;
Triangle T(n,k) begins as:
     1;
     3,       3;
     9,      42,        9;
    27,     393,      393,        27;
    81,    3156,     8646,      3156,        81;
   243,   23631,   142446,    142446,     23631,      243;
   729,  171006,  2015895,   4273380,   2015895,   171006,     729;
  2187, 1216725, 26107983, 102402705, 102402705, 26107983, 1216725, 2187;

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

Cf. A000407 (row sums), A142459, A257612.
Cf. A038221, A257180, A257611, A257620, A257623, A257625, A257627.
Similar sequences listed in A256890.

t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
T[n_, k_, p_, q_]= t[n-k, k, p, q];
Table[T[n,k,4,3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 01 2022 *)

@CachedFunction
def t(n,k,p,q):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
def A257621(n,k): return t(n-k,k,4,3)
flatten([[A257621(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2022

T(n, k) = t(n-k, k), where 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), and f(n) = 4*n + 3.
Sum_{k=0..n} T(n, k) = A000407(n).
From G. C. Greubel, Mar 01 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)

A257623 Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(n) = 5*n + 3.

Original entry on oeis.org

1, 3, 3, 9, 48, 9, 27, 501, 501, 27, 81, 4494, 13026, 4494, 81, 243, 37815, 250230, 250230, 37815, 243, 729, 309324, 4122735, 9008280, 4122735, 309324, 729, 2187, 2498649, 62256627, 256971945, 256971945, 62256627, 2498649, 2187

Offset: 0

Dale Gerdemann, May 10 2015

			Array, t(n,k), begins as:
    1,       3,         9,           27,             81, ... A000244;
    3,      48,       501,         4494,          37815, ...;
    9,     501,     13026,       250230,        4122735, ...;
   27,    4494,    250230,      9008280,      256971945, ...;
   81,   37815,   4122735,    256971945,    11820709470, ...;
  243,  309324,  62256627,   6368680566,   450199373658, ...;
  729, 2498649, 891791568, 144065371932, 15108742867890, ...;
Triangle, T(n,k), begins as:
     1;
     3,       3;
     9,      48,        9;
    27,     501,      501,        27;
    81,    4494,    13026,      4494,        81;
   243,   37815,   250230,    250230,     37815,      243;
   729,  309324,  4122735,   9008280,   4122735,   309324,     729;
  2187, 2498649, 62256627, 256971945, 256971945, 62256627, 2498649, 2187;

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

Cf. A000244, A008548, A142460, A257614.
Cf. A038221, A257180, A257611, A257620, A257621, A257625, A257627.
Similar sequences listed in A256890.

t[n_, k_, p_, q_]:= t[n, k, p, q]= If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+ q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
T[n_, k_, p_, q_]= t[n-k,k,p,q];
Table[T[n,k,5,3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 27 2022 *)

@CachedFunction
def t(n,k,p,q):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
def A257623(n,k): return t(n-k,k,5,3)
flatten([[A257623(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 27 2022

T(n,k) = t(n-k, k) where 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), and f(n) = 5*n + 3.
Sum_{k=0..n} T(n, k) = A008548(n).
From G. C. Greubel, Feb 27 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)

A257625 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 6*n + 3.

Original entry on oeis.org

1, 3, 3, 9, 54, 9, 27, 621, 621, 27, 81, 6156, 18630, 6156, 81, 243, 57591, 408726, 408726, 57591, 243, 729, 526338, 7685847, 17166492, 7685847, 526338, 729, 2187, 4765473, 132656859, 568014201, 568014201, 132656859, 4765473, 2187

Offset: 0

Dale Gerdemann, May 10 2015

			Array t(n,k) begins as:
    1,       3,          9,           27,             81, ...;
    3,      54,        621,         6156,          57591, ...;
    9,     621,      18630,       408726,        7685847, ...;
   27,    6156,     408726,     17166492,      568014201, ...;
   81,   57591,    7685847,    568014201,    30672766854, ...;
  243,  526338,  132656859,  16305974568,  1366261865802, ...;
  729, 4765473, 2175706332, 427278012876, 53552912878818, ...;
Triangle T(n,k) begins as:
     1;
     3,       3;
     9,      54,         9;
    27,     621,       621,        27;
    81,    6156,     18630,      6156,        81;
   243,   57591,    408726,    408726,     57591,       243;
   729,  526338,   7685847,  17166492,   7685847,    526338,     729;
  2187, 4765473, 132656859, 568014201, 568014201, 132656859, 4765473, 2187;

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

Cf. A047058 (row sums), A142461, A257616.
Cf. A038221, A257180, A257611, A257620, A257621, A257623, A257625, A257627.
See similar sequences listed in A256890.

t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
T[n_, k_, p_, q_]= t[n-k, k, p, q];
Table[T[n,k,6,3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 01 2022 *)

@CachedFunction
def t(n,k,p,q):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
def A257625(n,k): return t(n-k,k,6,3)
flatten([[A257625(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2022

T(n, k) = t(n-k, k), where 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), and f(n) = 6*n + 3.
Sum_{k=0..n} T(n, k) = A047058(n).
From G. C. Greubel, Mar 01 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)

A257627 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(x) = 7*x + 3.

Original entry on oeis.org

1, 3, 3, 9, 60, 9, 27, 753, 753, 27, 81, 8178, 25602, 8178, 81, 243, 84291, 631506, 631506, 84291, 243, 729, 852144, 13348623, 30312288, 13348623, 852144, 729, 2187, 8554245, 259308063, 1141302225, 1141302225, 259308063, 8554245, 2187

Offset: 0

Dale Gerdemann, May 10 2015

			Array t(n, k) begins as:
    1,       3,          9,            27,              81, ... A000244;
    3,      60,        753,          8178,           84291, ...;
    9,     753,      25602,        631506,        13348623, ...;
   27,    8178,     631506,      30312288,      1141302225, ...;
   81,   84291,   13348623,    1141302225,     70760737950, ...;
  243,  852144,  259308063,   37244959794,   3608891348622, ...;
  729, 8554245, 4793178096, 1109572049376, 161806374029202, ...;
Triangle, T(n, k) begins as:
     1;
     3,       3;
     9,      60,         9;
    27,     753,       753,         27;
    81,    8178,     25602,       8178,         81;
   243,   84291,    631506,     631506,      84291,       243;
   729,  852144,  13348623,   30312288,   13348623,    852144,     729;
  2187, 8554245, 259308063, 1141302225, 1141302225, 259308063, 8554245, 2187;

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

Cf. A000244, A038221, A049209 (row sums), A142462.
Cf. A257180, A257611, A257617, A257620, A257621, A257623, A257625.
See similar sequences listed in A256890.

f[n_]:= 7*n+3;
t[n_, k_]:= t[n,k]= If[n<0 || k<0, 0, If[n==0 && k==0, 1, f[k]*t[n-1,k] +f[n]*t[n,k-1]]];
T[n_, k_]= t[n-k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 22 2022 *)

def f(n): return 7*n+3
@CachedFunction
def t(n,k):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return f(k)*t(n-1, k) + f(n)*t(n, k-1)
def A257627(n,k): return t(n-k,k)
flatten([[A257627(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 22 2022

T(n, k) = t(n-k, k), where 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) = 7*x + 3.
Sum_{k=0..n} T(n, k) = A049209(n).
From G. C. Greubel, Feb 22 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)

A038220 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)2^j.

Original entry on oeis.org

1, 3, 2, 9, 12, 4, 27, 54, 36, 8, 81, 216, 216, 96, 16, 243, 810, 1080, 720, 240, 32, 729, 2916, 4860, 4320, 2160, 576, 64, 2187, 10206, 20412, 22680, 15120, 6048, 1344, 128, 6561, 34992, 81648, 108864, 90720, 48384, 16128, 3072, 256

Offset: 0

N. J. A. Sloane

Row sums give A000351; central terms give A119309. - Reinhard Zumkeller, May 14 2006

Triangle of coefficients in expansion of (3 + 2x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 23 2018

			Triangle begins:
   1;
   3,   2;
   9,  12,   4;
  27,  54,  36,   8;
  81, 216, 216,  96,  16;
  ...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A013620, A000079, A000244, A013613, A038221.

a038220 n k = a038220_tabl !! n !! k
a038220_row n = a038220_tabl !! n
a038220_tabl = iterate (\row ->
   zipWith (+) (map (* 3) (row ++ [0])) (map (* 2) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013, Apr 02 2011

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 3 t[n - 1, k] + 2 t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[ Expand[(3 + 2x)^n], x], {n, 0, 9}] // Flatten  (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[Binomial[i, j] *3^(i - j)*2^j, x], {i, 0, 9}, {j, 0, i}] // Flatten (* Zagros Lalo, Jul 23 2018 *)

T(i,j)=binomial(i,j)*3^(i-j)*2^j \\ Charles R Greathouse IV, Jul 19 2016

T(n,k) = A007318(n,k) * A036561(n,k). - Reinhard Zumkeller, May 14 2006
G.f.: 1/(1 - 3*x - 2*x*y). - Ilya Gutkovskiy, Apr 21 2017
T(0,0) = 1; T(n,k) = 3 T(n-1,k) + 2 T(n-1,k-1) for k = 0...n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 23 2018

A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

2, 3, 3, 5, 8, 5, 9, 18, 18, 9, 17, 40, 48, 40, 17, 33, 90, 120, 120, 90, 33, 65, 204, 300, 320, 300, 204, 65, 129, 462, 756, 840, 840, 756, 462, 129, 257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257, 513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

From G. C. Greubel, Jan 18 2025: (Start)
A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))*A007318(n, k). When (p, q) = (2, 1) this sequence is obtained.
Some related triangles are:
  (p, q) = (1, 1) : 2*A007318(n,k).
  (p, q) = (2, 2) : 2*A038208(n,k).
  (p, q) = (3, 2) : A154692(n,k).
  (p, q) = (3, 3) : 2*A038221(n,k). (End)

			Triangle begins as:
     2;
     3,    3;
     5,    8,     5;
     9,   18,    18,     9;
    17,   40,    48,    40,    17;
    33,   90,   120,   120,    90,    33;
    65,  204,   300,   320,   300,   204,    65;
   129,  462,   756,   840,   840,   756,   462,   129;
   257, 1040,  1904,  2240,  2240,  2240,  1904,  1040,   257;
   513, 2322,  4752,  6048,  6048,  6048,  6048,  4752,  2322,  513;
  1025, 5140, 11700, 16320, 16800, 16128, 16800, 16320, 11700, 5140, 1025;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2502, Fig. 3.

Cf. A000129, A001045, A025192, A038208, A038221, A069720, A107920, A154692.
Cf. A215149.
Sums include: A008776 (row), A010673 (alternating sign row).
Columns k: A000051 (k=0).
Main diagonal: A059304.

A154690:= func< n,k | (2^(n-k)+2^k)*Binomial(n,k) >;
[A154690(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

A154690 := proc(n,m) binomial(n,m)*(2^(n-m)+2^m) ; end proc: # R. J. Mathar, Jan 13 2011

T[n_, m_]:= (2^(n-m) + 2^m)*Binomial[n,m];
Table[T[n,m], {n,0,12}, {m,0,n}]//Flatten

from sage.all import *
def A154690(n,k): return (pow(2,n-k)+pow(2,k))*binomial(n,k)
print(flatten([[A154690(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

T(n, k) = (2^(n-k) + 2^k)*A007318(n, k).
Sum_{k=0..n} T(n, k) = A008776(n) = A025192(n+1).
From G. C. Greubel, Jan 18 2025: (Start)
T(n, n-k) = T(n, k) (symmetry).
T(n, 1) = n + A215149(n), n >= 1.
T(2*n-1, n) = 3*A069720(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000129(n+1) + A001045(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = n+1 + A107920(n+1). (End)

cp's OEIS Frontend

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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A257620 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 3*n + 3.

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A257611 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.

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A257180 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.

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A257621 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 4*n + 3.

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A257623 Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.

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A257625 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 6*n + 3.

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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.

A257620 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 3*n + 3.

A257611 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(n) = 2*n + 3.

A257180 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), and f(x) = x + 3.

A257621 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 4*n + 3.

A257623 Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(n) = 5*n + 3.

A257625 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 6*n + 3.

A257627 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(x) = 7*x + 3.

A038220 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)2^j.