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.

A269920 Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 0.

Original entry on oeis.org

1, 1, 1, 2, 5, 2, 5, 22, 22, 5, 14, 93, 164, 93, 14, 42, 386, 1030, 1030, 386, 42, 132, 1586, 5868, 8885, 5868, 1586, 132, 429, 6476, 31388, 65954, 65954, 31388, 6476, 429, 1430, 26333, 160648, 442610, 614404, 442610, 160648, 26333, 1430
Offset: 0

Views

Author

Gheorghe Coserea, Mar 14 2016

Keywords

Comments

Row n contains n+1 terms.

Examples

			Triangle starts:
n\f    [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[0]    1;
[1]    1,      1;
[2]    2,      5,      2;
[3]    5,      22,     22,     5;
[4]    14,     93,     164,    93,     14;
[5]    42,     386,    1030,   1030,   386,    42;
[6]    132,    1586,   5868,   8885,   5868,   1586,   132;
[7]    429,    6476,   31388,  65954,  65954,  31388,  6476,   429;
[8]    ...

Links

Crossrefs

Columns k=1-6 give: A000108, A000346, A000184, A000365, A000473, A000502.
Row sums give A000168 (column 0 of A269919).
Cf. A006294 (row maxima).

Programs

  • Mathematica
    Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n<0 || f<0 || g<0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n+1) ((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n - 1, f-1, g] + (2n-3) (2n-2) (2n-1)/12 Q[n-2, f, g-1] + 1/2 Sum[l = n-k; Sum[v = f-u; Sum[j = g-i; Boole[l >= 1 && v >= 1 && j >= 0] (2k-1) (2l-1) Q[k-1, u, i] Q[l-1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
Table[Q[n, f, 0], {n, 0, 8}, {f, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)
  • PARI
    N = 8; G = 0; gmax(n) = min(n\2, G);
Q = matrix(N + 1, N + 1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, length(Q)-1, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric('x);
concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))

A029887 A sum over scaled A000531 related to Catalan numbers C(n).

Original entry on oeis.org

1, 11, 82, 515, 2934, 15694, 80324, 397923, 1922510, 9105690, 42438076, 195165646, 887516252, 3997537980, 17857602568, 79200753059, 349051186494, 1529735010658, 6670733733260, 28959032959962, 125209652884756, 539384745200516, 2315840230811832, 9912689725127950
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Related to planar maps? - see A000184. - N. J. A. Sloane, Mar 11 2007

Links

Crossrefs

Cf. A000108, A000184, A000531, A002802.

Programs

  • Magma
    [(2*n+1)*(2*n+3)*(2*n+5)*Catalan(n)/3 - (n+2)*2^(2*n+1): n in [0..30]]; // Vincenzo Librandi, Mar 14 2014
  • Mathematica
    a[n_] := (2*n+1)*(2*n+3)*(2*n+5)*CatalanNumber[n]/3 - (n+2)*2^(2*n+1); Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Mar 12 2014 *)
CoefficientList[Series[(4 x - 1 + Sqrt[1 - 4 x])/(2 x (1 - 4 x)^3), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 14 2014 *)
  • SageMath
    [(n+2)*((n+3)*(n+4)*catalan_number(n+3) - 3*4^(n+2))//24 for n in range(31)] # G. C. Greubel, Jul 18 2024

Formula

a(n) = 4^n * Sum_{k=0..n} A000531(k+1)/4^k.
a(n) = (1/3)*(2*n+1)*(2*n+3)*(2*n+5)*Catalan(n) - (n+2)*2^(2*n+1).
a(n) = 4*a(n-1) + A000531(n+1).
G.f. c(x)/(1-4*x)^(5/2) = (2-c(x))/(1-4*x)^3, where c(x) = g.f. for Catalan numbers; also convolution of Catalan numbers with A002802.
G.f.: (4*x-1+sqrt(1-4*x))/(2*x*(1-4*x)^3). - Vincenzo Librandi, Mar 14 2014
From G. C. Greubel, Jul 18 2024: (Start)
a(n) = (1/24)*(n+2)*((n+3)*(n+4)*Catalan(n+3) - 3*4^(n+2)).
a(n) = (1/2)*A000184(n+2). (End)

Extensions

More terms from Vincenzo Librandi, Mar 14 2014

A270407 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus g.

Original entry on oeis.org

2, 22, 164, 70, 1030, 1720, 5868, 24164, 6468, 31388, 256116, 258972, 160648, 2278660, 5554188, 1169740, 795846, 17970784, 85421118, 66449432, 3845020, 129726760, 1059255456, 1955808460, 351683046, 18211380, 875029804, 11270290416, 40121261136, 26225260226
Offset: 2

Views

Author

Gheorghe Coserea, Mar 16 2016

Keywords

Comments

Row n contains floor(n/2) terms.

Examples

			Triangle starts:
n\g    [0]          [1]          [2]          [3]          [4]
[2]    2;
[3]    22;
[4]    164,         70;
[5]    1030,        1720;
[6]    5868,        24164,       6468;
[7]    31388,       256116,      258972;
[8]    160648,      2278660,     5554188,     1169740;
[9]    795846,      17970784,    85421118,    66449432;
[10]   3845020,     129726760,   1059255456,  1955808460,  351683046;
[11]   18211380,    875029804,   11270290416, 40121261136, 26225260226;
[12]   ...

Links

Crossrefs

Columns k=0-1 give: A000184, A006296.

Programs

  • Mathematica
    Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n+1) ((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n - 1, f-1, g] + (2n-3) (2n-2) (2n-1)/12 Q[n-2, f, g-1] + 1/2 Sum[l = n-k; Sum[v = f-u; Sum[j = g-i; Boole[l >= 1 && v >= 1 && j >= 0] (2k-1) (2l-1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
T[n_, g_] := Q[n, 3, g];
Table[T[n, g], {n, 2, 11}, {g, 0, Quotient[n, 2]-1}] // Flatten (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    N = 11; F = 3; gmax(n) = n\2;
Q = matrix(N + 1, N + 1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, length(Q)-1, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric('x + O('x^(F+1)));
concat(vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F))))
Showing 1-3 of 3 results.

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

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