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.

A167752 Hankel transform of A167750.

Original entry on oeis.org

1, 1, 1, 0, -1, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1
Offset: 0

Views

Author

Paul Barry, Nov 10 2009

Keywords

Comments

The nonzero terms appear to be indexed by the quarter-squares floor((n+1)^2/4) = A002620(n+1).
abs(a(n)) = A237347(n) - 2. - Reinhard Zumkeller, Mar 18 2014

Crossrefs

Cf. A167753.

A167749 Riordan array (1,xf(x)) where f(x) is the g.f. of A005169 (fountains of coins).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 2, 3, 3, 1, 0, 3, 6, 6, 4, 1, 0, 5, 11, 13, 10, 5, 1, 0, 9, 20, 27, 24, 15, 6, 1, 0, 15, 38, 54, 55, 40, 21, 7, 1, 0, 26, 70, 109, 120, 100, 62, 28, 8, 1, 0, 45, 129, 216, 258, 236, 168, 91, 36, 9, 1, 0, 78, 238, 423, 544, 540, 426, 266, 128, 45, 10, 1
Offset: 0

Views

Author

Paul Barry, Nov 10 2009

Keywords

Comments

Row sums are A167750. Diagonal sums are A167751.

Examples

			Triangle begins
1,
0, 1,
0, 1, 1,
0, 1, 2, 1,
0, 2, 3, 3, 1,
0, 3, 6, 6, 4, 1,
0, 5, 11, 13, 10, 5, 1,
0, 9, 20, 27, 24, 15, 6, 1,
0, 15, 38, 54, 55, 40, 21, 7, 1,
0, 26, 70, 109, 120, 100, 62, 28, 8, 1,
0, 45, 129, 216, 258, 236, 168, 91, 36, 9, 1,
0, 78, 238, 423, 544, 540, 426, 266, 128, 45, 10, 1,
0, 135, 437, 824, 1127, 1205, 1035, 721, 402, 174, 55, 11, 1

Formula

G.f.: 1/(1-xy/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-... (continued fraction).

A302015 Expansion of 1/(1 - x/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, -1, 0, 1, 0, -1, 1, 1, -2, -1, 2, 0, -2, 2, 3, -3, -3, 4, 0, -7, 3, 9, -5, -7, 10, 4, -17, -1, 21, -7, -21, 21, 19, -36, -13, 47, -5, -56, 36, 64, -69, -54, 104, 15, -147, 41, 177, -115, -168, 221, 116, -344, -15, 442, -159, -481, 422, 443, -736, -280, 1034, -90, -1276, 681
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 30 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A286509.
Cf. A007325, A167750, A302016.

Programs

  • Mathematica
    nmax = 68; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[1/(1 - x QPochhammer[x, x^5] QPochhammer[x^4, x^5]/(QPochhammer[x^2, x^5] QPochhammer[x^3, x^5])), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4))/((1 - x^(5*k-2))*(1 - x^(5*k-3)))).
a(0) = 1; a(n) = Sum_{k=1..n} A007325(k-1)*a(n-k).

A001404 Triangle of values of 2-d recurrence.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 2, 1, 9, 11, 5, 2, 1, 20, 25, 12, 5, 2, 1, 45, 57, 27, 12, 5, 2, 1, 102, 129, 62, 28, 12, 5, 2, 1, 231, 293, 141, 64, 28, 12, 5, 2, 1, 524, 665, 321, 146, 65, 28, 12, 5, 2, 1, 1189, 1510, 729, 333, 148, 65, 28, 12, 5, 2, 1, 2699, 3428, 1656
Offset: 0

Views

Author

N. J. A. Sloane [ I have temporarily mislaid the name of the person who sent this ]

Keywords

Comments

The first column of the triangle (see example) appears to be A167750. [Joerg Arndt, Jul 09 2012]

Examples

			Triangle starts
1,
1, 1,
2, 2, 1,
4, 5, 2, 1,
9, 11, 5, 2, 1,
20, 25, 12, 5, 2, 1,
45, 57, 27, 12, 5, 2, 1,
102, 129, 62, 28, 12, 5, 2, 1,
231, 293, 141, 64, 28, 12, 5, 2, 1,
524, 665, 321, 146, 65, 28, 12, 5, 2, 1,
1189, 1510, 729, 333, 148, 65, 28, 12, 5, 2, 1,

Crossrefs

Cf. A001410.

Programs

  • Maple
    a[ 0,0 ] := 1; for i from 1 to N do a[ i,0 ] := a[ i-1,0 ]+a[ i-1,1 ]; for j from 1 to i do a[ i,j ] := sum(a[ i-j,t ],t=0..min(j+1,N)) od; od;
  • PARI
    T(m,n)=if(mRalf Stephan */

Extensions

Sequence corrected by Sean A. Irvine, Jul 08 2012
Showing 1-4 of 4 results.

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

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