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.

Previous Showing 11-20 of 24 results. Next

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)

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)

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)

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)

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)

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

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

A257613 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 + 4.

Original entry on oeis.org

1, 4, 4, 16, 48, 16, 64, 416, 416, 64, 256, 3136, 6656, 3136, 256, 1024, 21888, 84608, 84608, 21888, 1024, 4096, 145664, 939520, 1692160, 939520, 145664, 4096, 16384, 939520, 9555456, 28195840, 28195840, 9555456, 939520, 16384, 65536, 5932032, 91475968, 415734784, 676700160, 415734784, 91475968, 5932032, 65536
Offset: 0

Views

Author

Dale Gerdemann, May 06 2015

Keywords

Examples

			Triangle begins as:
      1;
      4,      4;
     16,     48,      16;
     64,    416,     416,       64;
    256,   3136,    6656,     3136,      256;
   1024,  21888,   84608,    84608,    21888,    1024;
   4096, 145664,  939520,  1692160,   939520,  145664,   4096;
  16384, 939520, 9555456, 28195840, 28195840, 9555456, 939520, 16384;

Links

Crossrefs

Cf. A051580 (row sums), A060187, A257609, A257611, A257615.
Cf. A257606, A257622.
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,2,4], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
  • PARI
    f(x) = 2*x + 4;
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): # A257613
    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,2,4) 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) = 2*x + 4.
Sum_{k=0..n} T(n, k) = A051580(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 = 2, and b = 4. - G. C. Greubel, Mar 20 2022

A257615 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 + 5.

Original entry on oeis.org

1, 5, 5, 25, 70, 25, 125, 715, 715, 125, 625, 6380, 12870, 6380, 625, 3125, 52785, 186010, 186010, 52785, 3125, 15625, 416370, 2360295, 4092220, 2360295, 416370, 15625, 78125, 3180215, 27488205, 75698255, 75698255, 27488205, 3180215, 78125
Offset: 0

Views

Author

Dale Gerdemann, May 09 2015

Keywords

Examples

			Triangle begins as:
      1;
      5,       5;
     25,      70,       25;
    125,     715,      715,      125;
    625,    6380,    12870,     6380,      625;
   3125,   52785,   186010,   186010,    52785,     3125;
  15625,  416370,  2360295,  4092220,  2360295,   416370,   15625;
  78125, 3180215, 27488205, 75698255, 75698255, 27488205, 3180215, 78125;

Links

Crossrefs

Cf. A000351, A051582 (row sums), A060187, A257609, A257611, A257613.
Cf. A257607, A257624.
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,2,5], {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,2,5) 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 + 5.
Sum_{k=0..n} T(n, k) = A051582(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 = 5.
T(n, n-k) = T(n, k).
T(n, 0) = A000351(n).
T(n, 1) = 5*7^n - 5^n*(n+5). (End)

A257606 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) = x + 4.

Original entry on oeis.org

1, 4, 4, 16, 40, 16, 64, 296, 296, 64, 256, 1928, 3552, 1928, 256, 1024, 11688, 34808, 34808, 11688, 1024, 4096, 67656, 302352, 487312, 302352, 67656, 4096, 16384, 379240, 2423016, 5830000, 5830000, 2423016, 379240, 16384, 65536, 2076424, 18330496, 62617144, 93280000, 62617144, 18330496, 2076424, 65536
Offset: 0

Views

Author

Dale Gerdemann, May 03 2015

Keywords

Examples

			Triangle begins as:
      1;
      4,      4;
     16,     40,      16;
     64,    296,     296,      64;
    256,   1928,    3552,    1928,     256;
   1024,  11688,   34808,   34808,   11688,    1024;
   4096,  67656,  302352,  487312,  302352,   67656,   4096;
  16384, 379240, 2423016, 5830000, 5830000, 2423016, 379240, 16384;

Links

Crossrefs

Cf. A008292, A049388 (row sums), A256890, A257180, A257607.
Cf. A257613, A257622.
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,1,4], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
  • Sage
    def T(n,k,a,b): # A257606
    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,1,4) 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(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.
Sum_{k=0..n} T(n, k) = A049388(n).
T(n,0) = T(n,n) = 4^n. - Georg Fischer, Oct 02 2021
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 = 1, and b = 4.
T(n, n-k) = T(n, k).
T(n, 1) = 8*5^n - 4^n*(8+n).
T(n, 2) = 2*((56 +15*n +n^2)*4^(n-1) - 4*(8+n)*5^n + 3*6^(n+1)). (End)

Extensions

a(3) corrected by Georg Fischer, Oct 02 2021
Previous Showing 11-20 of 24 results. Next

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

Content is available under The OEIS End-User License Agreement.