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 1-5 of 5 results.

A154228 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 16, 1, 1, 47, 47, 1, 1, 103, 974, 103, 1, 1, 195, 5354, 5354, 195, 1, 1, 336, 19969, 147068, 19969, 336, 1, 1, 541, 60085, 1259253, 1259253, 60085, 541, 1, 1, 827, 156386, 7010503, 44432886, 7010503, 156386, 827, 1, 1, 1213, 365498, 30299614, 536255794, 536255794, 30299614, 365498, 1213, 1
Offset: 0

Views

Author

Roger L. Bagula, Jan 05 2009

Keywords

Comments

Row sums are: {1, 2, 18, 96, 1182, 11100, 187680, 2639760, 58768320, ...}.
The row sums of this class of sequences (see cross-references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = (n+1)*(n+2)*(2*n+3)/6 = A000330(n+1). - G. C. Greubel, Mar 02 2021

Examples

			Triangle begins as:
  1;
  1,    1;
  1,   16,      1;
  1,   47,     47,        1;
  1,  103,    974,      103,         1;
  1,  195,   5354,     5354,       195,         1;
  1,  336,  19969,   147068,     19969,       336,        1;
  1,  541,  60085,  1259253,   1259253,     60085,      541,      1;
  1,  827, 156386,  7010503,  44432886,   7010503,   156386,    827,    1;
  1, 1213, 365498, 30299614, 536255794, 536255794, 30299614, 365498, 1213, 1;

Links

Crossrefs

Cf. A154227, A154229, A154230, A154231, A154233.
Cf. A000330.

Programs

  • Magma
    f:= func< n | (n+1)*(n+2)*(2*n+3)/6 >;
function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021
  • Maple
    T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + ((n+1)*(n+2)*(2*n+3)/6)*T[n-2, k-1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)
  • Sage
    def f(n): return (n+1)*(n+2)*(2*n+3)/6
def T(n,k):
    if (k==0 or k==n): return 1
    else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

Extensions

Edited by G. C. Greubel, Mar 02 2021

A154229 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)^2*T(n-2, k-1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 38, 1, 1, 139, 139, 1, 1, 365, 8828, 365, 1, 1, 807, 70492, 70492, 807, 1, 1, 1592, 357459, 7062136, 357459, 1592, 1, 1, 2889, 1404923, 98777227, 98777227, 1404923, 2889, 1, 1, 4915, 4631612, 824036625, 14498379854, 824036625, 4631612, 4915, 1
Offset: 0

Views

Author

Roger L. Bagula, Jan 05 2009

Keywords

Comments

Row sums are: {1, 2, 40, 280, 9560, 142600, 7780240, 200370080, 16155726160, ...}.
The row sums of this class of sequences (see cross-references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = binomial(n+2, 2)^2 = A000537(n+1). - G. C. Greubel, Mar 02 2021

Examples

			Triangle begins as:
  1;
  1,    1;
  1,   38,       1;
  1,  139,     139,         1;
  1,  365,    8828,       365,           1;
  1,  807,   70492,     70492,         807,         1;
  1, 1592,  357459,   7062136,      357459,      1592,       1;
  1, 2889, 1404923,  98777227,    98777227,   1404923,    2889,    1;
  1, 4915, 4631612, 824036625, 14498379854, 824036625, 4631612, 4915, 1;

Links

Crossrefs

Cf. A154227, A154228, A154230, A154231, A154233.
Cf. A000537.

Programs

  • Magma
    f:= func< n | Binomial(n+2,2)^2 >;
function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021
  • Maple
    T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else T(n-1, k) + T(n-1, k-1) + binomial(n+2,2)^2*T(n-2, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + Binomial[n+2, 2]^2*T[n-2, k-1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)
  • Sage
    def f(n): return binomial(n+2,2)^2
def T(n,k):
    if (k==0 or k==n): return 1
    else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)^2*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

Extensions

Edited by G. C. Greubel, Mar 02 2021

A154230 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30)*T(n-2, k-1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 100, 1, 1, 455, 455, 1, 1, 1435, 98810, 1435, 1, 1, 3711, 1135370, 1135370, 3711, 1, 1, 8388, 7849141, 464306300, 7849141, 8388, 1, 1, 17161, 40410421, 10431621081, 10431621081, 40410421, 17161, 1, 1, 32495, 169040786, 130822910455, 7140071740062, 130822910455, 169040786, 32495, 1
Offset: 0

Views

Author

Roger L. Bagula, Jan 05 2009

Keywords

Comments

Row sums are: {1, 2, 102, 912, 101682, 2278164, 480021360, 20944097328, ...}.
The row sums of this class of sequences (see cross references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = (n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30 = A000538(n+1). - G. C. Greubel, Mar 02 2021

Examples

			Triangle begins as:
  1;
  1,     1;
  1,   100,        1;
  1,   455,      455,           1;
  1,  1435,    98810,        1435,           1;
  1,  3711,  1135370,     1135370,        3711,        1;
  1,  8388,  7849141,   464306300,     7849141,     8388,     1;
  1, 17161, 40410421, 10431621081, 10431621081, 40410421, 17161, 1;

Links

Crossrefs

Cf. A154227, A154228, A154229, A154231, A154233.
Cf. A000538.

Programs

  • Magma
    f:= func< n | (n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30 >;
function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021
  • Maple
    T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else T(n-1, k) +T(n-1, k-1) +((n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30)*T(n-2, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021
  • Mathematica
    T[n_, k_]:= T[n,k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + ((n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30)*T[n-2, k-1] ];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)
  • Sage
    def f(n): return (n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30
def T(n,k):
    if (k==0 or k==n): return 1
    else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

Extensions

Edited by G. C. Greubel, Mar 02 2021

A154231 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12)*T(n-2, k-1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 278, 1, 1, 1579, 1579, 1, 1, 6005, 1233308, 6005, 1, 1, 18207, 20504692, 20504692, 18207, 1, 1, 47216, 194715939, 35816807848, 194715939, 47216, 1, 1, 108993, 1319518787, 1302709376779, 1302709376779, 1319518787, 108993, 1, 1, 229819, 7024500980, 24830582225241, 4330171226988158, 24830582225241, 7024500980, 229819, 1
Offset: 0

Views

Author

Roger L. Bagula, Jan 05 2009

Keywords

Comments

Row sums are: {1, 2, 280, 3160, 1245320, 41045800, 36206334160, ...}.
The row sums of this class of sequences (see cross-references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = (n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12 = A000539(n+1). - G. C. Greubel, Mar 02 2021

Examples

			Triangle begins as:
  1;
  1,      1;
  1,    278,          1;
  1,   1579,       1579,             1;
  1,   6005,    1233308,          6005,             1;
  1,  18207,   20504692,      20504692,         18207,          1;
  1,  47216,  194715939,   35816807848,     194715939,      47216,      1;
  1, 108993, 1319518787, 1302709376779, 1302709376779, 1319518787, 108993, 1;

Links

Crossrefs

Cf. A154227, A154228, A154229, A154230, A154233.
Cf. A000539 (powers of 5).

Programs

  • Magma
    f:= func< n | Binomial(n+2,2)^2*(2*n^2+6*n+3)/3 >;
function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021
  • Maple
    T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2+6*n+3)/12)*T(n-2, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021
  • Mathematica
    T[n_, k_]:= T[n,k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + ((n+1)^2*(n+2)^2*(2*n^2+6*n+3)/12)*T[n-2, k-1] ];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)
  • Sage
    def f(n): return binomial(n+2,2)^2*(2*n^2+6*n+3)/3
def T(n,k):
    if (k==0 or k==n): return 1
    else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

Extensions

Edited by G. C. Greubel, Mar 02 2021

A154233 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 19, 19, 1, 1, 39, 171, 39, 1, 1, 69, 761, 761, 69, 1, 1, 111, 2429, 8533, 2429, 111, 1, 1, 167, 6335, 52817, 52817, 6335, 167, 1, 1, 239, 14383, 231611, 711477, 231611, 14383, 239, 1, 1, 329, 29485, 809809, 5643801, 5643801, 809809, 29485, 329, 1
Offset: 0

Views

Author

Roger L. Bagula, Jan 05 2009

Keywords

Comments

Row sums are: {1, 2, 9, 40, 251, 1662, 13615, 118640, 1203945, 12966850, ...}.
The row sums of this class of sequences (see cross references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = n^2 +n -1 = A028387(n-2). - G. C. Greubel, Mar 02 2021

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   7,     1;
  1,  19,    19,      1;
  1,  39,   171,     39,       1;
  1,  69,   761,    761,      69,       1;
  1, 111,  2429,   8533,    2429,     111,      1;
  1, 167,  6335,  52817,   52817,    6335,    167,     1;
  1, 239, 14383, 231611,  711477,  231611,  14383,   239,   1;
  1, 329, 29485, 809809, 5643801, 5643801, 809809, 29485, 329, 1;

Links

Crossrefs

Cf. A154227, A154228, A154229, A154230, A154231.
Cf. A028387.

Programs

  • Magma
    f:= func< n | n^2+n-1 >;
function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021
  • Maple
    T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else T(n-1, k) + T(n-1, k-1) + (n^2+n-1)*T(n-2, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021
  • Mathematica
    T[n_, k_]:= T[n,k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + (n^2+n-1)*T[n-2, k-1] ];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)
  • Sage
    def f(n): return n^2+n-1
def T(n,k):
    if (k==0 or k==n): return 1
    else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2+n-1)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

Extensions

Edited by G. C. Greubel, Mar 02 2021
Showing 1-5 of 5 results.

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