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.

A153199 Triangle read by rows, A007318 * A153198.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 20, 14, 1, 5, 20, 50, 70, 43, 1, 6, 30, 100, 210, 258, 143, 1, 7, 42, 175, 490, 903, 1001, 509, 1, 8, 56, 280, 980, 2408, 4004, 4072, 1922, 1, 9, 72, 420, 1764, 5418, 12012, 18324, 17298, 7651
Offset: 0

Views

Author

Gary W. Adamson, Dec 20 2008

Keywords

Comments

Row sums = A153197 starting (1, 2, 5, 15, 51, 189, 748, 3138,...).
Right border = A006789: (1, 1, 2, 5, 14, 43, 143, 509, 1922,...).

Examples

			First few rows of the triangle =
1;
1, 1;
1, 2, 2;
1, 3, 6, 5;
1, 4, 12, 20, 14;
1, 5, 20, 50, 70, 43;
1, 6, 30, 100, 210, 258, 143;
1, 7, 42, 175, 490, 903, 1001, 509;
1, 8, 56, 280, 980, 2408, 4004, 4072, 1922;
1, 9, 72, 420, 1764, 5418, 12012, 18324, 17298, 7651;
1, 10, 90, 600, 2940, 10836, 30030, 61080, 86490, 76510, 31965;
...

Crossrefs

Cf. A153197, A153198, A006789.

Formula

Binomial transform of triangle A153198, where A153198 = an infinite lower triangular matrix with the Bessel numbers A006789 as the main diagonal and the rest 0's.

A153206 Convolution triangle by rows, T(n,k) = A153197(n-k) * A153198.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 2, 2, 5, 15, 5, 4, 5, 14, 51, 15, 10, 10, 14, 43, 189, 51, 30, 25, 28, 43, 143, 748, 189, 102, 75, 70, 86, 143, 509, 3128, 748, 378, 255, 210, 215, 286, 509, 1922, 13731, 3128, 1496, 945, 714, 645, 715, 1018, 1922, 7651
Offset: 0

Views

Author

Gary W. Adamson, Dec 20 2008

Keywords

Comments

A153197 prefaced with a 1: (1, 1, 2, 5, 15, 51,...) convolved with A006789 (1, 1, 2, 5, 14, 43,...) = A006789 shifted: (1, 2, 5, 14, 43, 143,...).
Right border = A006789, row sums = A006789 shifted.

Examples

			First few rows of the triangle =
1;
1, 1;
2, 1, 2;
5, 2, 2, 5;
15, 5, 4, 5, 14;
51, 15, 10, 10, 14, 43;
189, 51, 30, 25, 28, 43, 143;
748, 189, 102, 75, 70, 86, 143, 509;
3128, 748, 378, 255, 210, 215, 286, 509, 1922;
13731, 3128, 1496, 945, 714, 645, 715, 1018, 1922, 7651;
62969, 13731, 6256, 3740, 2646, 2193, 2145, 2545, 3844, 7651, 31965;
...
Row 5 = (51, 15, 10, 10, 14, 43), = termwise products of (51, 15, 5, 2, 1, 1) and (1, 1, 2, 5, 14, 43), where A153197 = (1, 2, 5, 15, 51,...); and A006789 = (1, 1, 2, 5, 14, 43,...).

Crossrefs

Cf. A006789, A153197

Formula

Convolution triangle by rows, T(n,k) = A153197(n-k) * A153198 = a * b, where a = an infinite lower triangular matrix with A153197 prefaced with a 1: (1, 1, 2, 5, 15, 51, 189, 748,...) in every column; and b = an infinite lower triangular matrix with A006789 in the main diagonal and the rest zeros.

A153197 Binomial transform of Bessel numbers A006789.

Original entry on oeis.org

1, 2, 5, 15, 51, 189, 748, 3128, 13731, 62969, 300552, 1488704, 7634723, 40464741, 221311617, 1247444859, 7238458309, 43196661875, 264878725516, 1667564565616, 10770316016557, 71314258947903, 483765644021787, 3359905164274725
Offset: 0

Views

Author

Gary W. Adamson, Dec 20 2008

Keywords

Comments

Prefaced with a 1: (1, 1, 2, 5, 15, 51, ...), convolved with A006789 = A006789; identical to reversing k terms of one sequence then taking the dot product of k terms of the other: e.g., A006789(6) = 143 = (51, 15, 5, 2, 1, 1) dot (1, 1, 2, 5, 14, 43) = (51 + 15 + 10 + 10 + 14 + 43). Equals row sums of triangle A153199. A153197 can be generated from the Hankel transform [1,1,1,...] by taking successive iterates of the operations: (binomial transform of [1,1,1,...]) followed by INVERT transform of the result, then binomial transform of the result, (repeat cycle)...; until the operations converge upon a two sequence fixed limit cycle of A006789 and A153197. Or, the infinite set of operations Q may begin: INVERT transform of [1,1,1,..] followed by binomial transform of the result, INVERT transform of the result, etc; until the operations again converge upon A006789 and A153197. The two sequences A006789 and A153197 have the mutual relationships that binomial transform of A006789 = A153197; while the INVERT transform of A153197 prefaced with a 1 and then prefaced with a 1 afterward = A006789.
Product of the two sequences (A153197 prefaced with a 1) and A006789 = A006789 with offset 1. Or, (1,1,2,5,15,51,...) * (1,1,2,5,14,43,...) = (1,2,5,14,43,...).
Conjecture: Given any sequence with Hankel transform of [1,1,1,...], performing alternate operations: binomial transform followed by INVERT transform, then binomial transform of the last result (repeat); or INVERT transform starting first, will converge upon A006789 and A153197 as a two sequence limit cycle. The conjecture can be extended to any Hankel transform (and their accompanying sequence set): analogous operations will converge upon a Bessel-type sequence and its binomial transform.

Examples

			a(3) = 15 = (1, 3, 3, 1) dot (1, 1, 2, 5) = (1 + 3 + 6 + 5).
G.f. = 1 + 2*x + 5*x^2 + 15*x^3 + 51*x^4 + 189*x^5 + 748*x^6 + 3128*x^7 + ...

Links

Crossrefs

Cf. A153198, A153199, A006789.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[{m = n + 1}, SeriesCoefficient[ Nest[ Series[ x / (1 - 2 x - x (Normal[#] /. x -> x / (1 - x))), {x, 0, m}] &, 0, Ceiling[m/2]], {x, 0, m}]]]; (* Michael Somos, Aug 05 2014 *)
  • PARI
    {a(n) = local( A = O(x) ); if( n<0, 0, n++; for( k=1, ceil(n/2), A = x / (1 - 2*x - x * subst( A, x, x / (1 - x)))); polcoeff( A, n))}; /* Michael Somos, Mar 06 2011 */

Formula

INVERT transform of A153197 with 1 prepended is A006789 with initial 1 removed. Bessel numbers with offset 1: (1, 2, 5, 14, 43, 143, ...).
If the offset is 1, the g.f. A(x) satisfies A(x) = x / (1 - 2*x - x * A( x / (1 - x))). - Michael Somos, Mar 06 2011
G.f.: 1/U(0) where U(k)= 1 - x*(k+2) - x^2/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: 1/x^2-1/x-U(0)/x^2 where U(k)= 1 - x*(k+1) - x^2/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 24 2012

Extensions

Edited by N. J. A. Sloane, Mar 06 2011
Showing 1-3 of 3 results.

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

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