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.

Previous Showing 41-45 of 45 results.

A352006 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 2^k * a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 21, 59, 117, 283, 597, 1467, 3125, 7387, 16149, 39931, 87541, 207643, 463061, 1107515, 2473909, 5819739, 13132437, 31080571, 70236533, 164315035, 373572693, 875121339, 1991869237, 4639482331, 10599986709, 24765957371, 56617082101
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 28 2022

Keywords

Crossrefs

Cf. A000621, A015083, A352007, A352008.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[2^k a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 32}]
nmax = 32; A[] = 0; Do[A[x] = 1/(1 - x A[2 x^2]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(2*x^2)).

A376111 a(0) = 1; a(n) = Sum_{k=1..n} (2^k-1) * a(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 4, 35, 600, 19942, 1299768, 167796051, 43131308656, 22127283690338, 22680691426392504, 46472849736334410494, 190399379929624643874384, 1559942353285454499773312748, 25559656412925984160985399396784, 837564388804449970974724247002202883
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2024

Keywords

Crossrefs

Cf. A000556, A015083, A376112, A376113.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(2^k - 1) a[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
nmax = 15; A[] = 0; Do[A[x] = 1/(1 + x A[x] - 2 x A[2 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 + x * A(x) - 2 * x * A(2*x)).

A154829 A q-Catalan triangle for q=2.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 17, 25, 9, 1, 171, 258, 102, 16, 1, 3113, 4635, 1788, 290, 25, 1, 106419, 154048, 54909, 7910, 665, 36, 1, 7035649, 9907933, 3232971, 385669, 26257, 1323, 49, 1, 915028347, 1262093470, 382948336, 37703584, 1889650, 71596, 2380, 64, 1
Offset: 0

Views

Author

Paul Barry, Jan 15 2009

Keywords

Comments

First column is A015083. Row sums are A154828.

Examples

			Triangle begins
1,
1, 1,
3, 4, 1,
17, 25, 9, 1,
171, 258, 102, 16, 1,
3113, 4635, 1788, 290, 25, 1

Crossrefs

Cf. A060693.

Formula

Triangle [1,2,4,8,16,32,...] DELTA [1,0,1,0,1,0,1,....] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 28 2011
G.f.: 1/(1-(x+xy)/(1-2x/(1-(4x+xy)/(1-8x/(1-(16x+xy)/(1-.... (continued fraction).

A376923 T(n, k) = T(n - 1, k) + 2^(n - 1)*T(n - 2, k - 1), if k > 0 and T(n, 0) = 2^n.

Original entry on oeis.org

1, 2, 0, 4, 2, 0, 8, 10, 0, 0, 16, 42, 16, 0, 0, 32, 170, 176, 0, 0, 0, 64, 682, 1520, 512, 0, 0, 0, 128, 2730, 12400, 11776, 0, 0, 0, 0, 256, 10922, 99696, 206336, 65536, 0, 0, 0, 0, 512, 43690, 798576, 3380736, 3080192, 0, 0, 0, 0, 0, 1024, 174762, 6390640, 54425088, 108724224, 33554432, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Thomas Scheuerle, Oct 17 2024

Keywords

Comments

This is the case r = 2 of the more general recurrence: T(n, k, r) = T(n-1, k, r) + r^(n-1)*T(n - 2, k - 1, r), if k > 0 and T(n, 0, r) = 1 + (r^n - 1)/(r - 1) if r > 1. Consider the sequence b(n) = Sum_{k=0..n-1} b(n - k - 1)*T(n - 1, k, r)*(-1)^k, with b(0) = 1. The sequence b(n) will have an ordinary generating function which can be represented as the continued fraction expansion: 1/(1 - x/(1 - r^0*x/(1 - r^1*x/(1 - r^2*x/(1 - r^3*x/(...)))))). In short b(n) will have the ordinary generating function 1/(1-G(x)*x), where G(x) is the generating function of the Carlitz-Riordan q-Catalan numbers for q = r. The Hankel determinant of b(0)..b(2*n) will be r^A016061(n). The Hankel determinant of b(1)..b(2*n+1) will be r^A002412(n).

Examples

			Triangle begins:
n\k  0 |  1 |  2 | 3 | 4 | 5
[0]  1,
[1]  2,   0
[2]  4,   2,   0
[3]  8,  10,   0,  0
[4] 16,  42,  16,  0,  0
[5] 32, 170, 176,  0,  0,  0

Crossrefs

Cf. A002412, A015083, A016061, A377132.

Programs

  • PARI
    T(n, k) = if(n < 0, return(0), return(if(k == 0, return(2^n), T(n-1,k) + 2^(n-1)*T(n-2,k-1))))

Formula

Column k has o.g.f.: x^(2*k)*2^(k^2)/((1 - 2^(k+1)*x)*Product_{m=1..k}(1 - 2^(m-1)*x)).

A385618 G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(2*x) + A(3*x)) ).

Original entry on oeis.org

1, 2, 14, 230, 9014, 913334, 254986934, 203241812630, 471322195238102, 3214892041613961206, 64937611960188470964662, 3901256965326759127330935830, 699101347969640933511109922382422, 374397435055450676411068538643233721206, 599979003238812649083869782544110463986119734
Offset: 0

Views

Author

Seiichi Manyama, Jul 05 2025

Keywords

Crossrefs

Cf. A007689, A015083, A015084, A047749, A385617.

Programs

  • Mathematica
    terms = 15; A[] = 1; Do[A[x] = 1/( 1 - x*(A[2*x] + A[3*x]) ) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 05 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (2^j+3^j)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (2^k+3^k) * a(k) * a(n-1-k).
Previous Showing 41-45 of 45 results.

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