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 1-3 of 3 results.

A157011 Triangle T(n,k) read by rows: T(n,k)= (k-1)*T(n-1,k) + (n-k+2)*T(n-1, k-1), with T(n,1)=1, for 1 <= k <= n, n >= 1.

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 23, 8, 1, 14, 82, 93, 16, 1, 20, 234, 607, 343, 32, 1, 27, 588, 2991, 3800, 1189, 64, 1, 35, 1365, 12501, 30155, 21145, 3951, 128, 1, 44, 3010, 47058, 195626, 256500, 108286, 12749, 256, 1, 54, 6416, 165254, 1111910, 2456256, 1932216, 522387, 40295, 512
Offset: 1

Views

Author

Roger L. Bagula, Feb 21 2009

Keywords

Comments

Row sums are apparently in A002627.
The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - G. C. Greubel, Feb 22 2019

Examples

			The triangle starts in row n=1 as:
  1;
  1,  2;
  1,  5,    4;
  1,  9,   23,      8;
  1, 14,   82,     93,      16;
  1, 20,  234,    607,     343,      32;
  1, 27,  588,   2991,    3800,    1189,      64;
  1, 35, 1365,  12501,   30155,   21145,    3951,    128;
  1, 44, 3010,  47058,  195626,  256500,  108286,  12749,   256;
  1, 54, 6416, 165254, 1111910, 2456256, 1932216, 522387, 40295, 512;

Links

Crossrefs

Cf. A000096 (column k=1), A002627, A008517.
Cf. This sequence (m=0), A008292 (m=1), A157012 (m=2), A157013 (m=3).

Programs

  • Maple
    A157011 := proc(n,k) if k <0 or k >= n then 0; elif k  =0 then 1; else k*procname(n-1,k)+(n-k+1)*procname(n-1,k-1) ; end if; end proc: # R. J. Mathar, Jun 18 2011
  • Mathematica
    e[n_, 0, m_]:= 1;
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
T[n_, 1]:= 1; T[n_, n_]:= 2^(n-1); T[n_, k_]:= T[n, k] = (k-1)*T[n-1, k] + (n-k+2)*T[n-1, k-1]; Table[T[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)
  • PARI
    {T(n, k) = if(k==1, 1, if(k==n, 2^(n-1), (k-1)*T(n-1, k) + (n-k+2)* T(n-1, k-1)))};
for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019
  • Sage
    def T(n, k):
    if (k==1):
        return 1
    elif (k==n):
        return 2^(n-1)
    else: return (k-1)*T(n-1, k) + (n-k+2)* T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Feb 22 2019

A157012 Riordan's general Eulerian recursion: T(n,k) = (k+2)*T(n-1, k) + (n-k) * T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 18, 14, 1, 0, 1, 58, 110, 33, 1, 0, 1, 179, 672, 495, 72, 1, 0, 1, 543, 3583, 5163, 1917, 151, 1, 0, 1, 1636, 17590, 43730, 32154, 6808, 310, 1, 0, 1, 4916, 81812, 324190, 411574, 176272, 22904, 629, 1, 0
Offset: 0

Views

Author

Roger L. Bagula, Feb 21 2009

Keywords

Comments

Row sums are:
{1, 1, 2, 7, 34, 203, 1420, 11359, 102230, 1022299,...}.
This recursion set doesn't seem to produce the Eulerian 2nd A008517.
The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - G. C. Greubel, Feb 22 2019

Examples

			Triangle begins with:
1.
1,    0.
1,    1,     0.
1,    5,     1,      0.
1,   18,    14,      1,      0.
1,   58,   110,     33,      1,      0.
1,  179,   672,    495,     72,      1,     0.
1,  543,  3583,   5163,   1917,    151,     1,   0.
1, 1636, 17590,  43730,  32154,   6808,   310,   1,   0.
1, 4916, 81812, 324190, 411574, 176272, 22904, 629,   1,   0.

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215

Links

Crossrefs

Cf. A008517.
Cf. A157011 (m=0), A008292 (m=1), This Sequence (m=2), A157013 (m=3).

Programs

  • Mathematica
    e[n_, 0, m_]:= 1;
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] +(n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
T[n_, 0]:= 1; T[n_, n_]:= 0; T[n_, k_]:= T[n, k] = (k+2)*T[n-1, k] +(n-k) *T[n-1, k-1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)
  • PARI
    {T(n, k) = if(k==0, 1, if(k==n, 0, (k+2)*T(n-1, k) + (n-k)* T(n-1, k-1)))};
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019
  • Sage
    def T(n, k):
    if (k==0): return 1
    elif (k==n): return 0
    else: return (k+2)*T(n-1, k) + (n-k)* T(n-1, k-1)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 22 2019

Formula

e(n,k,m) = (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) for m=2.
T(n,k) = (k+2)*T(n-1, k) + (n-k)*T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0. - G. C. Greubel, Feb 22 2019

A306547 Triangle read by rows, defined by Riordan's general Eulerian recursion: T(n, k) = (k+3)*T(n-1, k) + (n-k-2) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).

Original entry on oeis.org

1, 1, -2, 1, -11, 4, 1, -55, 35, -8, 1, -274, 210, -91, 16, 1, -1368, 986, -637, 219, -32, 1, -6837, 3180, -3473, 1752, -507, 64, 1, -34181, -1431, -17951, 10543, -4563, 1147, -128, 1, -170900, -145310, -129950, 48442, -30524, 11470, -2555, 256, 1, -854494, -1726360, -1490890, -2314, -177832, 84176, -28105, 5627, -512
Offset: 1

Views

Author

G. C. Greubel, Feb 22 2019

Keywords

Comments

Row sums are {1, -1, -6, -27, -138, -831, -5820, -46563, -419070, -4190703, ...}.
The Mathematica code for e(n,k,m) gives eleven sequences of which the first few are in the OEIS (see Crossrefs section).

Examples

			Triangle begins with:
1.
1,      -2.
1,     -11,       4.
1,     -55,      35,      -8.
1,    -274,     210,     -91,    16.
1,   -1368,     986,    -637,   219,    -32.
1,   -6837,    3180,   -3473,  1752,   -507,    64.
1,  -34181,   -1431,  -17951, 10543,  -4563,  1147,  -128.
1, -170900, -145310, -129950, 48442, -30524, 11470, -2555, 256.

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215.

Links

Crossrefs

Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), A157013 (m=3), this sequence.

Programs

  • Mathematica
    e[n_, 0, m_]:= 1; (* Example for m=3 *)
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
T[n_, 1]:= 1; T[n_, n_]:= (-2)^(n-1); T[n_, k_]:= T[n, k] = (k+3)*T[n-1, k] + (n-k-2)*T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]//Flatten
  • PARI
    {T(n, k) = if(k==1, 1, if(k==n, (-2)^(n-1), (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)))};
for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")))
  • Sage
    def T(n, k):
    if (k==1): return 1
    elif (k==n): return (-2)^(n-1)
    else: return (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..12)]

Formula

T(n, k) = (k+3)*T(n-1, k) + (n-k-2)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).
e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.
Showing 1-3 of 3 results.

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

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