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 1-4 of 4 results.

A144643 Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3 or 4 (n >= 0, 0 <= k <= 4n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 0, 1, 3, 7, 15, 25, 35, 35, 0, 0, 0, 1, 6, 25, 90, 280, 770, 1855, 3675, 5775, 5775, 0, 0, 0, 0, 1, 10, 65, 350, 1645, 6930, 26425, 90475, 275275, 725725, 1576575, 2627625, 2627625, 0, 0, 0, 0, 0, 1, 15, 140, 1050, 6825, 39795, 211750, 1033725, 4629625
Offset: 0

Views

Author

David Applegate and N. J. A. Sloane, Jan 25 2009

Keywords

Examples

			Irregular triangle begins:
  1;
  0, 1, 1, 1, 1;
  0, 0, 1, 3, 7, 15, 25,  35,  35;
  0, 0, 0, 1, 6, 25, 90, 280, 770, 1855, 3675, 5775, 5775;
  ...

Links

Crossrefs

Row sums give A144508.
See A144644 and A144645 for other versions.
Cf. A144299, A144385.

Programs

  • Magma
    function t(n,k)
  if k eq n then return 1;
  elif k le n-1 or n le 0 then return 0;
  else return (&+[Binomial(k-1,j)*t(n-1,k-j-1): j in [0..3]]);
  end if;
end function;
A144643:= func< n,k | t(n,k) >;
[A144643(n,k): k in [0..4*n], n in [0..8]]; // G. C. Greubel, Oct 11 2023
  • Maple
    T := proc(n, k) option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else T(n - 1, k - 1) + (k - 1)*T(n - 1, k - 2) + 1/2*(k - 1)*(k - 2)*T(n - 1, k - 3) + 1/6*(k - 1)*(k - 2)*(k - 3)*T(n - 1, k - 4);
end if;
end proc;
  • Mathematica
    T[n_, k_]:= T[n, k]= Which[n==k, 1, kJean-François Alcover, Mar 20 2014, after Maple *)
Table[BellY[k, n, {1,1,1,1}], {n,0,12}, {k,0,4*n}]//Flatten (* G. C. Greubel, Oct 11 2023 *)
  • SageMath
    @CachedFunction
def t(n,k):
    if (k==n): return 1
    elif (kA144643(n,k): return t(n,k)
flatten([[A144643(n,k) for k in range(4*n+1)] for n in range(13)]) # G. C. Greubel, Oct 11 2023

Formula

T(n, k) = Sum_{j=0..3} binomial(k-1, j) * T(n-1, k-j-1), with T(n, n) = 1, T(n, k) = 0 if n < 1 or n > k.
Sum_{k=0..4*n} T(n, k) = A144508(n).

A151511 The triangle in A151359 read by rows downwards.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 1, 31, 90, 65, 15, 1, 0, 0, 63, 301, 350, 140, 21, 1, 0, 0, 119, 966, 1701, 1050, 266, 28, 1, 0, 0, 210, 2989, 7770, 6951, 2646, 462, 36, 1, 0, 0, 336, 8925, 33985, 42525, 22827, 5880, 750, 45, 1, 0, 0, 462, 25641
Offset: 0

Views

Author

N. J. A. Sloane, May 14 2009

Keywords

Comments

The Bell transform of g(n) = 1 if n<6 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Examples

			Triangle begins:
1
0 1
0 1 1
0 1 3 1
0 1 7 6 1
0 1 15 25 10 1
0 1 31 90 65 15 1
0 0 63 301 350 140 21 1
0 0 119 966 1701 1050 266 28 1

Links

Crossrefs

Begins in same way as triangle of Stirling numbers of second kind, A048993, but is strictly different. N. J. A. Sloane, Aug 09 2017
Cf. A148092 (row sums), A122848, A111246, A144644, A151509.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; a[n_ /; 1 <= n <= 6] = 1; a[] = 0; T[n, k_] := T[n, k] = If[k == 0, a[0]^n, Sum[Binomial[n - 1, j - 1] a[j] T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 20 2016, after Peter Luschny *)
  • Sage
    # uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<6 else 0, 12) # Peter Luschny, Jan 19 2016

Formula

Bivariate e.g.f. A151511(x,t) = Sum_{n>=0, k>=0} T(n,k)*x^n*t^k/n! = exp(t*G6(x)), where G6(x) = Sum_{i=1..6} x^i/i! is the e.g.f. of column 1. - R. J. Mathar, May 28 2019

Extensions

Row 9 added by Michel Marcus, Feb 13 2014
More rows from R. J. Mathar, May 28 2019

A151509 The triangle in A151338 read by rows downwards.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 0, 31, 90, 65, 15, 1, 0, 0, 56, 301, 350, 140, 21, 1, 0, 0, 91, 938, 1701, 1050, 266, 28, 1, 0, 0, 126, 2737, 7686, 6951, 2646, 462, 36, 1, 0, 0, 126, 7455, 32725, 42315, 22827, 5880, 750, 45, 1, 0, 0, 0, 18711, 132055
Offset: 0

Views

Author

N. J. A. Sloane, May 14 2009

Keywords

Comments

The Bell transform of the sequence "g(n) = 1 if n < 5, otherwise 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,   1;
  0, 1,  7,   6,    1;
  0, 1, 15,  25,   10,    1;
  0, 0, 31,  90,   65,   15,   1;
  0, 0, 56, 301,  350,  140,  21,  1;
  0, 0, 91, 938, 1701, 1050, 266, 28, 1;

Links

Crossrefs

Cf. A110038 (row sums), A122848, A111246, A144644, A151511.

Programs

  • Mathematica
    rows = 10;
BellMatrix[f_Function | f_Symbol, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[If[# < 5, 1, 0]&, rows];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)
  • Sage
    # uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<5 else 0, 12) # Peter Luschny, Jan 19 2016

Formula

Bivariate e.g.f A151509(x,t) = Sum_{n>=0, k>=0} T(n,k)*x^n*t^k/n! = exp(t*G5(x)), where G5(x) = Sum_{i=1..5} x^i/i! is the e.g.f. of column 1. - R. J. Mathar, May 28 2019

Extensions

Row 9 added by Michel Marcus, Feb 13 2014
More rows from R. J. Mathar, May 28 2019

A144645 Triangle in A144643 read upwards by columns.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 0, 0, 1, 15, 65, 90, 25, 0, 0, 1, 21, 140, 350, 280, 35, 0, 0, 1, 28, 266, 1050, 1645, 770, 35, 0, 0, 1, 36, 462, 2646, 6825, 6930, 1855, 0, 0, 0, 1, 45, 750, 5880, 22575, 39795, 26425, 3675, 0, 0, 0
Offset: 0

Views

Author

David Applegate and N. J. A. Sloane, Jan 25 2009

Keywords

Examples

			Triangle begins:
  1;
  1,  0;
  1,  1,   0;
  1,  3,   1,    0;
  1,  6,   7,    1,     0;
  1, 10,  25,   15,     0,     0;
  1, 15,  65,   90,    25,     0,     0;
  1, 21, 140,  350,   280,    35,     0,    0;
  1, 28, 266, 1050,  1645,   770,    35,    0,   0;
  1, 36, 462, 2646,  6825,  6930,  1855,    0,   0,   0;
  1, 45, 750, 5880, 22575, 39795, 26425, 3675,   0,   0,   0;

Links

Crossrefs

Cf. A001681 (row sums), A144643, A144644.

Programs

  • Magma
    function t(n,k)
  if k eq n then return 1;
  elif k le n-1 or n le 0 then return 0;
  else return (&+[Binomial(k-1,j)*t(n-1,k-j-1): j in [0..3]]);
  end if;
end function;
A144645:= func< n,k | t(n-k,n) >;
[A144645(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 11 2023
  • Mathematica
    Table[BellY[n, n-k, {1,1,1,1}], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 11 2023; based on A144644 *)
  • SageMath
    @CachedFunction
def t(n,k):
    if (k==n): return 1
    elif (kA144645(n,k): return t(n-k,n)
flatten([[A144645(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 11 2023

Formula

From G. C. Greubel, Oct 11 2023: (Start)
T(n, k) = A144643(n-k, n).
T(n, k) = A144644(n, n-k).
T(n, k) = t(n-k, n), where t(n, k) = Sum_{j=0..3} binomial(k-1, j) * t(n-1, k-j-1), with t(n,n) = 1, t(n,k) = 0 if n < 1 or n > k.
Sum_{k=0..n} T(n, k) = A001681(n). (End)
Showing 1-4 of 4 results.

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

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