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.

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

A151510 The triangle in A151338 read by rows upwards.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, May 14 2009

Keywords

Comments

Columns 1-5 are A000012, A000217, A001296, A001297, and A001298. - Nathaniel Johnston, Apr 30 2011

Examples

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

Extensions

Extended by Nathaniel Johnston, Apr 30 2011

A151359 Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3, 4, 5 or 6 (n >= 0, 0 <= k <= 6n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 3, 7, 15, 31, 63, 119, 210, 336, 462, 462, 0, 0, 0, 1, 6, 25, 90, 301, 966, 2989, 8925, 25641, 70455, 183183, 441441, 966966, 1849848, 2858856, 2858856, 0, 0, 0, 0, 1, 10, 65, 350, 1701, 7770, 33985, 143605, 588511
Offset: 0

Views

Author

N. J. A. Sloane, May 14 2009

Keywords

Comments

Row n has 6n+1 entries.

Examples

			Triangle begins:
[0, 1, 1, 1, 1, 1, 1]
[0, 0, 1, 3, 7, 15, 31, 63, 119, 210, 336, 462, 462]
[0, 0, 0, 1, 6, 25, 90, 301, 966, 2989, 8925, 25641, 70455, 183183, 441441, 966966, 1849848, 2858856, 2858856]
[0, 0, 0, 0, 1, 10, 65, 350, 1701, 7770, 33985, 143605, 588511, 2341339, 9032023, 33668635, 120681561, 413104692, 1337944608, 4046710668, 11216721516, 27756632904, 58555088592, 96197645544, 96197645544]
[0, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42525, 246400, 1370985, 7383376, 38657619, 197212015, 980839860, 4752728981, 22399494117, 102410296989, 452572985865, 1924000439361, 7820764020069, 30157961878044, 109184327692440, 365935843649376, 1113006758944080, 2982608000091720, 6696799094545560, 11423951396577720, 11423951396577720]
...

Links

Crossrefs

This is one of a sequence of triangles: A144331, A144385, A144643, A151338, A151359, ...
See A151511, A151512 for other versions.

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}]]; T[n_, k_] := t[k, n+1]; Table[Table[T[n, k], {k, 0, 6(n+1)} ], {n, 0, 4}] // Flatten (* Jean-François Alcover, Jan 20 2016, using Peter Luschny's Bell transform *)
Showing 1-3 of 3 results.

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

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