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.

User: Dale Gerdemann

Dale Gerdemann's wiki page.

Dale Gerdemann has authored 25 sequences. Here are the ten most recent ones:

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)

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

Views

Author

Dale Gerdemann, May 10 2015

Keywords

Examples

			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;

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    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

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), 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)

A257626 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 + 6.

Original entry on oeis.org

1, 6, 6, 36, 108, 36, 216, 1404, 1404, 216, 1296, 15876, 33696, 15876, 1296, 7776, 166212, 642492, 642492, 166212, 7776, 46656, 1659204, 10701720, 19274760, 10701720, 1659204, 46656, 279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936
Offset: 0

Views

Author

Dale Gerdemann, May 10 2015

Keywords

Examples

			Triangle begins as:
       1;
       6,        6;
      36,      108,        36;
     216,     1404,      1404,       216;
    1296,    15876,     33696,     15876,      1296;
    7776,   166212,    642492,    642492,    166212,      7776;
   46656,  1659204,  10701720,  19274760,  10701720,   1659204,    46656;
  279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936;

Links

Crossrefs

Cf. A051609 (row sums), A142458, A257610, A257620, A257622, A257624.
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,3,6], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
  • Sage
    def T(n,k,a,b): # A257626
    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,3,6) 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 + 6.
Sum_{k=0..n} T(n, k) = A051609(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 = 6. - G. C. Greubel, Mar 20 2022

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

Views

Author

Dale Gerdemann, May 10 2015

Keywords

Examples

			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;

Links

Crossrefs

Cf. A047058 (row sums), A142461, A257616.
Cf. A038221, A257180, A257611, A257620, A257621, A257623, A257625, A257627.
See 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,6,3], {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 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

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) = 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)

A257624 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 + 5.

Original entry on oeis.org

1, 5, 5, 25, 80, 25, 125, 915, 915, 125, 625, 9070, 20130, 9070, 625, 3125, 83185, 348410, 348410, 83185, 3125, 15625, 727980, 5246655, 9755480, 5246655, 727980, 15625, 78125, 6183215, 72272805, 225769855, 225769855, 72272805, 6183215, 78125
Offset: 0

Views

Author

Dale Gerdemann, May 10 2015

Keywords

Examples

			Triangle begins as:
      1;
      5,       5;
     25,      80,       25;
    125,     915,      915,       125;
    625,    9070,    20130,      9070,       625;
   3125,   83185,   348410,    348410,     83185,     3125;
  15625,  727980,  5246655,   9755480,   5246655,   727980,   15625;
  78125, 6183215, 72272805, 225769855, 225769855, 72272805, 6183215, 78125;

Links

Crossrefs

Cf. A051607 (row sums), A142458, A257610, A257620, A257622, A257626.
Cf. A257607, A257615.
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,3,5], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
  • Sage
    def T(n,k,a,b): # A257624
    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,3,5) 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 + 5.
Sum_{k=0..n} T(n, k) = A051607(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 = 5. - G. C. Greubel, Mar 20 2022

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

Views

Author

Dale Gerdemann, May 10 2015

Keywords

Examples

			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;

Links

Crossrefs

Cf. A000244, A008548, A142460, A257614.
Cf. A038221, A257180, A257611, A257620, A257621, 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,5,3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 27 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 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

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 + 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)

A257622 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 + 4.

Original entry on oeis.org

1, 4, 4, 16, 56, 16, 64, 552, 552, 64, 256, 4696, 11040, 4696, 256, 1024, 36968, 171448, 171448, 36968, 1024, 4096, 278232, 2305968, 4457648, 2305968, 278232, 4096, 16384, 2037736, 28346088, 94844912, 94844912, 28346088, 2037736, 16384
Offset: 0

Views

Author

Dale Gerdemann, May 10 2015

Keywords

Examples

			Triangle begins as:
      1;
      4,       4;
     16,      56,       16;
     64,     552,      552,       64;
    256,    4696,    11040,     4696,      256;
   1024,   36968,   171448,   171448,    36968,     1024;
   4096,  278232,  2305968,  4457648,  2305968,   278232,    4096;
  16384, 2037736, 28346088, 94844912, 94844912, 28346088, 2037736, 16384;

Links

Crossrefs

Cf. A051605 (row sums), A142458, A257610, A257620, A257624, A257626.
Cf. A257606, A257613.
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,3,4], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
  • Sage
    def T(n,k,a,b): # A257622
    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,3,4) 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 + 4.
Sum_{k=0..n} T(n, k) = A051605(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 = 4. - G. C. Greubel, Mar 20 2022

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

Views

Author

Dale Gerdemann, May 09 2015

Keywords

Examples

			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;

Links

Crossrefs

Cf. A000407 (row sums), A142459, A257612.
Cf. A038221, A257180, A257611, A257620, 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,4,3], {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 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

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) = 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)

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

A257618 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) = 8*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 40, 4, 8, 472, 472, 8, 16, 4928, 16992, 4928, 16, 32, 49824, 433984, 433984, 49824, 32, 64, 499584, 9505728, 22567168, 9505728, 499584, 64, 128, 4999040, 192085632, 909941120, 909941120, 192085632, 4999040, 128
Offset: 0

Views

Author

Dale Gerdemann, May 09 2015

Keywords

Examples

			Triangle begins as:
    1;
    2,       2;
    4,      40,         4;
    8,     472,       472,         8;
   16,    4928,     16992,      4928,        16;
   32,   49824,    433984,    433984,     49824,        32;
   64,  499584,   9505728,  22567168,   9505728,    499584,      64;
  128, 4999040, 192085632, 909941120, 909941120, 192085632, 4999040, 128;

Links

Crossrefs

Cf. A000079, A144828 (row sums), A167884.
Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257617, 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,8,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
  • Sage
    def T(n,k,a,b): # A257618
    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,8,2) 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) = 8*x + 2.
Sum_{k=0..n} T(n, k) = A144828(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 = 8, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = 2^(n-1)*(5^n - 2*n - 1).
T(n, 2) = 2^(n-3)*(3^(2*n+1) -2*(2*n+1)*5^n -1 +4*n^2). (End)

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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