cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 24 results. Next

A257609 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) = 2*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 16, 4, 8, 88, 88, 8, 16, 416, 1056, 416, 16, 32, 1824, 9664, 9664, 1824, 32, 64, 7680, 76224, 154624, 76224, 7680, 64, 128, 31616, 549504, 1999232, 1999232, 549504, 31616, 128, 256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256
Offset: 0

Views

Author

Dale Gerdemann, May 03 2015

Keywords

Examples

			Triangle begins as:
    1;
    2,      2;
    4,     16,       4;
    8,     88,      88,        8;
   16,    416,    1056,      416,       16;
   32,   1824,    9664,     9664,     1824,       32;
   64,   7680,   76224,   154624,    76224,     7680,      64;
  128,  31616,  549504,  1999232,  1999232,   549504,   31616,    128;
  256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256;

Links

Crossrefs

Cf. A000079, A002866 (row sums), A008292, A060187, A100575, A257611, A257613.
Cf. A257615, A038208, A256890, A257610, A257612, A257614, A257616, A257617.
Cf. A257618, A257619.
Similar sequences listed in A256890.

Programs

  • Magma
    function T(n,k,a,b)
  if k lt 0 or k gt n then return 0;
  elif k eq 0 or k eq n then return 1;
  else return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b);
  end if; return T;
end function;
[T(n,k,2,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 21 2022
  • Magma
    A257609:= func< n,k | 2^n*EulerianNumber(n+1, k) >;
[A257609(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 17 2025
  • Mathematica
    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,2,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)
  • Sage
    def T(n,k,a,b): # A257609
    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,2,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

Formula

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) = 2*x + 2.
Sum_{k=0..n} T(n, k) = A002866(n).
From G. C. Greubel, Mar 21 2022: (Start)
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 = 2, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = 2*A100575(n+1). (End)
T(n, k) = 2^n*A008292(n+1, k+1). - G. C. Greubel, Jan 17 2025

A257610 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) = 3*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 20, 4, 8, 132, 132, 8, 16, 748, 2112, 748, 16, 32, 3964, 25124, 25124, 3964, 32, 64, 20364, 256488, 552728, 256488, 20364, 64, 128, 103100, 2398092, 9670840, 9670840, 2398092, 103100, 128, 256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256
Offset: 0

Views

Author

Dale Gerdemann, May 03 2015

Keywords

Examples

			Triangle begins as:
    1;
    2,      2;
    4,     20,        4;
    8,    132,      132,         8;
   16,    748,     2112,       748,        16;
   32,   3964,    25124,     25124,      3964,        32;
   64,  20364,   256488,    552728,    256488,     20364,       64;
  128, 103100,  2398092,   9670840,   9670840,   2398092,   103100,    128;
  256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256;

Links

Crossrefs

Cf. A007559 (row sums), A038208, A142458, A257620, A257622, A257624, A257626.
Cf. A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257618, A257619.
See similar sequences listed in A256890.

Programs

  • Mathematica
    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,3,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
  • Sage
    def T(n,k,a,b): # A257610
    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,3,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

Formula

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) = 3*x + 2.
Sum_{k=0..n} T(n, k) = A007559(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 = 3, and b = 2. - G. C. Greubel, Mar 20 2022

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

Views

Author

Dale Gerdemann, May 09 2015

Keywords

Examples

			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;

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • Python
    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
  • Sage
    @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

Formula

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

Views

Author

Dale Gerdemann, May 06 2015

Keywords

Examples

			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;

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    @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

Formula

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

Views

Author

Dale Gerdemann, Apr 17 2015

Keywords

Examples

			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;

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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)

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

Views

Author

Dale Gerdemann, May 06 2015

Keywords

Comments

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

Examples

			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;

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

A257614 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) = 5*n + 2.

Original entry on oeis.org

1, 2, 2, 4, 28, 4, 8, 244, 244, 8, 16, 1844, 5856, 1844, 16, 32, 13260, 101620, 101620, 13260, 32, 64, 93684, 1511160, 3455080, 1511160, 93684, 64, 128, 657836, 20663388, 91981880, 91981880, 20663388, 657836, 128, 256, 4609588, 269011408, 2121603436, 4047202720, 2121603436, 269011408, 4609588, 256
Offset: 0

Views

Author

Dale Gerdemann, May 09 2015

Keywords

Examples

			Array t(n,k) begins as:
   1,      2,         4,           8,            16, ... A000079;
   2,     28,       244,        1844,         13260, ...;
   4,    244,      5856,      101620,       1511160, ...;
   8,   1844,    101620,     3455080,      91981880, ...;
  16,  13260,   1511160,    91981880,    4047202720, ...;
  32,  93684,  20663388,  2121603436,  146321752612, ...;
  64, 657836, 269011408, 44675623468, 4648698508440, ...;
Triangle T(n,k) begins as:
    1;
    2,      2;
    4,     28,        4;
    8,    244,      244,        8;
   16,   1844,     5856,     1844,       16;
   32,  13260,   101620,   101620,    13260,       32;
   64,  93684,  1511160,  3455080,  1511160,    93684,     64;
  128, 657836, 20663388, 91981880, 91981880, 20663388, 657836, 128;

Links

Crossrefs

Cf. A000079, A008546 (row sums), A142460, A257623.
Cf. A038208, A256890, A257609, A257610, A257612, A257616, A257617, A257618, A257619.
Similar sequences listed in A256890.

Programs

  • Mathematica
    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,5,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 01 2022 *)
  • Sage
    @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 A257614(n,k): return t(n-k,k,5,2)
flatten([[A257614(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2022

Formula

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 + 2.
Sum_{k=0..n} T(n, k) = A008546(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) = A000079(n). (End)

A257616 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) = 6*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 32, 4, 8, 312, 312, 8, 16, 2656, 8736, 2656, 16, 32, 21664, 175424, 175424, 21664, 32, 64, 174336, 3019200, 7016960, 3019200, 174336, 64, 128, 1397120, 47847552, 218838400, 218838400, 47847552, 1397120, 128, 256, 11182592, 722956288, 5907889664, 11379596800, 5907889664, 722956288, 11182592, 256
Offset: 0

Views

Author

Dale Gerdemann, May 09 2015

Keywords

Examples

			Triangle begins as:
    1;
    2,       2;
    4,      32,        4;
    8,     312,      312,         8;
   16,    2656,     8736,      2656,        16;
   32,   21664,   175424,    175424,     21664,       32;
   64,  174336,  3019200,   7016960,   3019200,   174336,      64;
  128, 1397120, 47847552, 218838400, 218838400, 47847552, 1397120, 128;

Links

Crossrefs

Cf. A000079, A049308 (row sums), A142461, A257625.
Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257617, A257618, A257619.
Similar sequences listed in A256890.

Programs

  • Mathematica
    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,6,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)
  • Sage
    def T(n,k,a,b): # A257610
    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,6,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

Formula

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) = 6*x + 2.
Sum_{k=0..n} T(n, k) = A049308(n).
From G. C. Greubel, Mar 21 2022: (Start)
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 = 6, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = (2^n/3)*(2^(2*n+1) - (3*n+2)). (End)

A257617 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) = 7*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 36, 4, 8, 388, 388, 8, 16, 3676, 12416, 3676, 16, 32, 33564, 283204, 283204, 33564, 32, 64, 303260, 5538184, 13027384, 5538184, 303260, 64, 128, 2732156, 99831564, 465775352, 465775352, 99831564, 2732156, 128
Offset: 0

Views

Author

Dale Gerdemann, May 09 2015

Keywords

Examples

			    1;
    2,       2;
    4,      36,        4;
    8,     388,      388,         8;
   16,    3676,    12416,      3676,        16;
   32,   33564,   283204,    283204,     33564,       32;
   64,  303260,  5538184,  13027384,   5538184,   303260,      64;
  128, 2732156, 99831564, 465775352, 465775352, 99831564, 2732156, 128;

Links

Crossrefs

Cf. A000079, A142462, A144827 (row sums), A257627.
Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257618, A257619
Similar sequences listed in A256890.

Programs

  • Mathematica
    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,7,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
  • Sage
    def T(n,k,a,b): # A257617
    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,7,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

Formula

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) = 7*x + 2.
Sum_{k=0..n} T(n, k) = A144827(n).
From G. C. Greubel, Mar 24 2022: (Start)
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 = 7, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = (4*9^n - 2^n*(7*n + 4))/7.
T(n, 2) = (2^(n-1)*(49*n^2 +7*n -12) + 11*2^(4*n+1) - 4*(7*n+4)*9^n)/49. (End)

A257619 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) = 9*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 44, 4, 8, 564, 564, 8, 16, 6436, 22560, 6436, 16, 32, 71404, 637844, 637844, 71404, 32, 64, 786948, 15470232, 36994952, 15470232, 786948, 64, 128, 8660012, 346391196, 1660722424, 1660722424, 346391196, 8660012, 128
Offset: 0

Views

Author

Dale Gerdemann, May 09 2015

Keywords

Examples

			Triangle begins as:
    1;
    2,       2;
    4,      44,         4;
    8,     564,       564,          8;
   16,    6436,     22560,       6436,         16;
   32,   71404,    637844,     637844,      71404,        32;
   64,  786948,  15470232,   36994952,   15470232,    786948,      64;
  128, 8660012, 346391196, 1660722424, 1660722424, 346391196, 8660012, 128;

Links

Crossrefs

Cf. A000079, A144829 (row sums), A257608.
Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257618.
Similar sequences listed in A256890.

Programs

  • Mathematica
    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,9,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
  • PARI
    f(x) = 9*x + 2;
t(n, m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, 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, k), ", "); ); print(); ); } \\ Michel Marcus, May 23 2015
  • Sage
    def T(n,k,a,b): # A257619
    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,9,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

Formula

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) = 9*x + 2.
Sum_{k=0..n} T(n, k) = A144829(n).
From G. C. Greubel, Mar 24 2022: (Start)
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 = 9, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = (1/9)*(4*11^n - 2^n*(9*n + 4)).
T(n, 2) = (1/81)*(26*20^n - 4*(4+9*n)*11^n - 2^(n-1)*(20 + 9*n - 81*n^2)). (End)
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