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.

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A288071 a(n) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus 1.

Original entry on oeis.org

420, 14065, 256116, 3392843, 36703824, 344468530, 2908358552, 22620890127, 164767964504, 1137369687454, 7506901051000, 47700234551918, 293370096957504, 1753945289216484, 10229201477344752, 58364244137596695, 326571194881454376, 1795631576981016038, 9718877491130813368, 51858415558095569962
Offset: 5

Views

Author

Gheorghe Coserea, Jun 05 2017

Keywords

Links

Crossrefs

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, this sequence, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
Column 4 of A269921.
Cf. A000108.

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) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 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] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 4, 1];
Table[a[n], {n, 5, 24}] (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288071_ser(N) = {
  my(y = A000108_ser(N+1));
  y*(y-1)^5*(307*y^4 + 2411*y^3 - 2094*y^2 - 708*y + 504)/(y-2)^14;
};
Vec(A288071_ser(20))

A288072 a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 1.

Original entry on oeis.org

2310, 100156, 2278660, 36703824, 472592916, 5188948072, 50534154408, 448035881592, 3682811916980, 28442316247080, 208462422428152, 1461307573813824, 9857665477085832, 64309102366765200, 407372683115470800, 2514120288996270024, 15159074541052024308, 89512241718624419624
Offset: 6

Views

Author

Gheorghe Coserea, Jun 05 2017

Keywords

Links

Crossrefs

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, this sequence, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
Column 5 of A269921.
Cf. A000108.

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) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 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] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 5, 1];
Table[a[n], {n, 6, 23}] (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288072_ser(N) = {
  my(y = A000108_ser(N+1));
  -2*y*(y-1)^6*(2140*y^5 + 14751*y^4 - 15604*y^3 - 8820*y^2 + 10176*y - 1488)/(y-2)^17;
};
Vec(A288072_ser(18))

A288073 a(n) is the number of rooted maps with n edges and 9 faces on an orientable surface of genus 1.

Original entry on oeis.org

1385670, 126264820, 5593305476, 164767964504, 3682811916980, 67173739068760, 1046677747672360, 14373136466094880, 177882700353757460, 2017523504473479992, 21241931655650633720, 209732362862241103248, 1957830216739337392584, 17394726697224718134384, 147908195064869691109072
Offset: 10

Views

Author

Gheorghe Coserea, Jun 05 2017

Keywords

Links

Crossrefs

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, this sequence, A288074 f=10.
Column 9 of A269921.

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) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 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] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 9, 1];
Table[a[n], {n, 10, 24}] (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288073_ser(N) = {
  my(y = A000108_ser(N+1));
  -2*y*(y-1)^10*(58911256*y^9 + 315266323*y^8 - 563073084*y^7 - 706445836*y^6 + 1588166368*y^5 - 488205920*y^4 - 472512192*y^3 + 315108288*y^2 - 44342784*y - 2179584)/(y-2)^29;
};
Vec(A288073_ser(17))

A288074 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 1.

Original entry on oeis.org

6466460, 678405090, 34225196720, 1137369687454, 28442316247080, 576218752277476, 9908748651241088, 149314477245194262, 2017523504473479992, 24868664942648145372, 283389619978690157408, 3017066587822315930220, 30265092793614787511376, 288055728071446557904968, 2616366012933033221518720
Offset: 11

Views

Author

Gheorghe Coserea, Jun 05 2017

Keywords

Links

Crossrefs

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, this sequence.
Column 10 of A269921.

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) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 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] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 10, 1];
Table[a[n], {n, 11, 25}] (* Jean-François Alcover, Oct 18 2018 *)
  • PARI
    A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288074_ser(N) = {
  my(y = A000108_ser(N+1));
  2*y*(y-1)^11*(734641583*y^10 + 3795452665*y^9 - 7483071778*y^8 - 10235465624*y^7 + 25178445968*y^6 - 7563355856*y^5 - 11624244832*y^4 + 8854962048*y^3 - 1433163264*y^2 - 286758144*y + 65790464)/(y-2)^32;
};
Vec(A288074_ser(15))

A343092 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without isthmuses, n >= 2, k = 1..n-1.

Original entry on oeis.org

1, 4, 10, 10, 79, 70, 20, 340, 900, 420, 35, 1071, 5846, 7885, 2310, 56, 2772, 26320, 71372, 59080, 12012, 84, 6258, 93436, 431739, 706068, 398846, 60060, 120, 12768, 280120, 2000280, 5494896, 6052840, 2499096, 291720, 165, 24090, 739420, 7643265, 32055391, 58677420, 46759630, 14805705, 1385670
Offset: 2

Views

Author

Andrew Howroyd, Apr 04 2021

Keywords

Comments

The number of vertices is n - k.
Column k is a polynomial of degree 3*k. This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.

Examples

			Triangle begins:
   1;
   4,   10;
  10,   79,    70;
  20,  340,   900,    420;
  35, 1071,  5846,   7885,   2310;
  56, 2772, 26320,  71372,  59080,  12012;
  84, 6258, 93436, 431739, 706068, 398846, 60060;
  ...

Links

Crossrefs

Columns 1..2 are A000292, A006469.
Diagonals are A002802, A006425, A006426, A006427.
Row sums are A343093.
Cf. A269921, A342981, A342989, A343090.

Programs

  • PARI
    \\ Needs F from A342989.
G(n,m,y,z)={my(p=F(n,m,y,z)); subst(p, x, serreverse(x*p^2))}
H(n, g=1)={my(q=G(n, g, 'y, 'z)-x, v=Vec(polcoef(sqrt(serreverse(x/q^2)/x), g, 'y))); [Vecrev(t) | t<-v]}
{ my(T=H(10)); for(n=1, #T, print(T[n])) }

A020922 Expansion of 1/(1-4*x)^(11/2).

Original entry on oeis.org

1, 22, 286, 2860, 24310, 184756, 1293292, 8498776, 53117350, 318704100, 1848483780, 10418726760, 57302997180, 308554600200, 1630931458200, 8480843582640, 43464323361030, 219878341708740, 1099391708543700, 5439095821216200, 26651569523959380, 129450480544945560
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also convolution of A000984 with A040075, also convolution of A000302 with A020920, also convolution of A002457 with A038846, also convolution of A002697 with A020918, also convolution of A002802 with A038845. - Rui Duarte, Oct 08 2011

Links

Crossrefs

Cf. A000302, A000984, A001622, A002457, A002697, A002802, A020918, A020920, A038845, A038846, A040075, A046521 (sixth column).

Programs

  • GAP
    List([0..30], n-> Binomial(n+5, 5)*Binomial(2*n+10, n+5)/252); # G. C. Greubel, Jul 20 2019
  • Magma
    [(2*n+9)*(2*n+7)*(2*n+5)*(2*n+3)*(2*n+1)*Binomial(2*n, n)/945: n in [0..30]] // Vincenzo Librandi, Jul 05 2013
  • Mathematica
    CoefficientList[Series[1/(1-4x)^(11/2), {x,0,30}], x] (* Vincenzo Librandi, Jul 05 2013 *)
  • PARI
    vector(30, n, n--; m=n+5; binomial(m, 5)*binomial(2*m, m)/252) \\ G. C. Greubel, Jul 20 2019
  • Sage
    [binomial(n+5, 5)*binomial(2*n+10, n+5)/252 for n in (0..30)] # G. C. Greubel, Jul 20 2019

Formula

a(n) = binomial(n+5, 5)*A000984(n+5)/A000984(5), where A000984 are central binomial coefficients. - Wolfdieter Lang
From Rui Duarte, Oct 08 2011: (Start)
a(n) = ((2n+9)(2n+7)(2n+5)(2n+3)(2n+1)/(9*7*5*3*1)) * binomial(2n, n).
a(n) = binomial(2n+10, 10) * binomial(2n, n) / binomial(n+5, 5).
a(n) = binomial(n+5, 5) * binomial(2n+10, n+5) / binomial(10, 5).
a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10+i_11 = n } f(i_1)* f(i_2)*f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10)*f(i_11) with f(k)=A000984(k). (End)
Boas-Buck recurrence: a(n) = (22/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n+5, 5). See a comment there. - Wolfdieter Lang, Aug 10 2017
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 162*sqrt(3)*Pi - 30816/35.
Sum_{n>=0} (-1)^n/a(n) = 4500*sqrt(5)*log(phi) - 33888/7, where phi is the golden ratio (A001622). (End)

A241269 Denominator of c(n) = (n^2+n+2)/((n+1)*(n+2)*(n+3)).

Original entry on oeis.org

3, 6, 15, 60, 105, 21, 126, 360, 495, 330, 429, 1092, 1365, 420, 1020, 2448, 2907, 1710, 1995, 4620, 5313, 759, 3450, 7800, 8775, 4914, 5481, 12180, 13485, 3720, 8184, 17952, 19635, 10710, 11655, 25308, 27417, 3705, 15990, 34440, 37023, 19866, 21285, 45540
Offset: 0

Views

Author

Paul Curtz, Apr 18 2014

Keywords

Comments

All terms are multiples of 3.
Difference table of c(n):
1/3, 1/6, 2/15, 7/60, 2/21,...
-1/6, -1/30, -1/60, -1/84, -1/105,...
2/15, 1/60, 1/210, 1/420, 1/630,...
-7/60, -1/84, -1/420, -1/1260, -1/2520,... .
This is an autosequence of the second kind; the inverse binomial transform is the signed sequence. The main diagonal is the first upper diagonal multiplied by 2.
Denominators of the main diagonal: A051133(n+1).
Denominators of the first upper diagonal; A000911(n).
c(n) is a companion to A026741(n)/A045896(n).
Based on the Akiyama-Tanigawa transform applied to 1/(n+1) which yields the Bernoulli numbers A164555(n)/A027642(n).
Are the numerators of the main diagonal (-1)^n? If yes, what is the value of 1/3 - 1/30 + 1/210,... or 1 - 1/10 + 1/70 - 1/420, ... , from A002802(n)?
Is a(n+40) - a(n) divisible by 10?
No: a(5) = 21 but a(45) = 12972. # Robert Israel, Jul 17 2023
Are the common divisors to A014206(n) and A007531(n+3) of period 16: repeat 2, 4, 4, 2, 2, 16, 4, 2, 2, 4, 4, 2, 2, 8, 4, 2?
Reduce c(n) = f(n) = b(n)/a(n) = 1/3, 1/6, 2/15, 7/60, 11/105, 2/21, 11/126, 29/360, ... .
Consider the successively interleaved autosequences (also called eigensequences) of the second kind and of the first kind
1, 1/2, 1/3, 1/4, 1/5, 1/6, ...
0, 1/6, 1/6, 3/20, 2/15, 5/42, ...
1/3, 1/6, 2/15, 7/60, 11/105, 2/21, ...
0, 1/10, 1/10, 13/140, 3/35, 5/63, ...
1/5, 1/10, 3/35, 11/140, 23/315, 43/630, ...
0, 1/14, 1/14, 17/252, 4/63, ...
This array is Au1(m,n). Au1(0,0)=1, Au1(0,1)=1/2.
Au1(m+1,n) = 2*Au1(m,n+1) - Au1(m,n).
First row: see A003506, Leibniz's Harmonic Triangle.
Second row: A026741/A045896.
a(n) is the denominator of the third row f(n).
The first column is 1, 0, 1/3, 0, 1/5, 0, 1/7, 0, ... . Numerators: A093178(n+1). This incites, considering tan(1), to introduce before the first row
Ta0(n) = 0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, ... .

Links

Programs

  • Maple
    seq(denom((n^2+n+2)/((n+1)*(n+2)*(n+3))),n=0..1000);
  • Mathematica
    Denominator[Table[(n^2+n+2)/Times@@(n+{1,2,3}),{n,0,50}]] (* Harvey P. Dale, Mar 27 2015 *)
  • PARI
    for(n=0, 100, print1(denominator((n^2+n+2)/((n+1)*(n+2)*(n+3))), ", ")) \\ Colin Barker, Apr 18 2014

Formula

c(n) = A014206(n)/A007531(n+3).
The sum of the difference table main diagonal is 1/3 - 1/30 + 1/210 - ... = 10*A086466-4 = 4*(sqrt(5)*log(phi)-1) = 0.3040894... - Jean-François Alcover, Apr 22 2014
a(n) = (n+1)*(n+2)*(n+3)/gcd(4*n - 4, n^2 + n + 2), where gcd(4*n - 4, n^2 + n + 2) is periodic with period 16. - Robert Israel, Jul 17 2023

Extensions

More terms from Colin Barker, Apr 18 2014

A051133 a(n) = binomial(2n,n)*n*(2n+1)/2.

Original entry on oeis.org

0, 3, 30, 210, 1260, 6930, 36036, 180180, 875160, 4157010, 19399380, 89237148, 405623400, 1825305300, 8143669800, 36064823400, 158685222960, 694247850450, 3022020054900, 13095420237900, 56517076816200, 243023430309660, 1041528987041400
Offset: 0

Views

Author

Wouter Meeussen

Keywords

Examples

			G.f. = 3*x + 30*x^2 + 210*x^3 + 1260*x^4 + 6930*x^5 + 36036*x^6 + ...

Links

Crossrefs

Cf. A000911, A000984, A000217, A002802.

Programs

  • Magma
    [Binomial(2*n,n)*n*(2*n+1)/2: n in [0..25]]; // G. C. Greubel, Feb 10 2019
  • Maple
    seq(binomial(2*n,n)*binomial(n,(n-2))/2, n=1..23); # Zerinvary Lajos, May 05 2007
  • Mathematica
    a[ n_]:= SeriesCoefficient[ 3x(1-4x)^(-5/2), {x, 0, n}]; (* Michael Somos, Sep 09 2013 *)
Table[Binomial[2*n, n]*n*(2*n + 1)/2, {n, 0, 22}] (* Amiram Eldar, Oct 22 2020 *)
  • PARI
    {a(n) = if( n<1, 0, (2*n + 1)! / (2 * n! *(n-1)!))}; /* Michael Somos, Sep 09 2013 */
  • PARI
    {a(n) = 2^(n+2) * polcoeff( pollegendre( n+3), n-1)}; /* Michael Somos, Sep 09 2013 */
  • Sage
    [binomial(2*n,n)*n*(2*n+1)/2 for n in (0..25)] # G. C. Greubel, Feb 10 2019

Formula

a(n) = (1/2) * A000911(n-1).
a(n) = (1/2)*A000984(n+1)*A000217(n). - Zerinvary Lajos, May 05 2007
a(n) = 3*A002802(n-1). - Zerinvary Lajos, Jun 02 2007
(-n+1)*a(n) + 2*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013
G.f.: 3*x * (1 - 4*x)^(-5/2). - Michael Somos, Sep 09 2013
Sum_{n>=1} 1/a(n) = 4 - 2*Pi/sqrt(3). - Amiram Eldar, Oct 22 2020

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

A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

Original entry on oeis.org

1, 24, 336, 3584, 32256, 258048, 1892352, 12976128, 84344832, 524812288, 3148873728, 18320719872, 103817412608, 574988746752, 3121367482368, 16647293239296, 87398289506304, 452414675091456, 2312341672689664, 11683410556747776, 58417052783738880, 289303499500421120
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Also convolution of A020922 with A000984 (central binomial coefficients); also convolution of A040075 with A000302 (powers of 4).
With a different offset, number of n-permutations of 5 objects: u,v,z,x, y with repetition allowed, containing exactly five (5) u's. Example: a(1)=24 because we have uuuuuv uuuuvu uuuvuu uuvuuu uvuuuu vuuuuu uuuuuz uuuuzu uuuzuu uuzuuu uzuuuu zuuuuu uuuuux uuuuxu uuuxuu uuxuuu uxuuuu xuuuuu uuuuuy uuuuyu uuuyuu uuyuuu uyuuuu yuuuuu. - Zerinvary Lajos, Jun 16 2008
Also convolution of A002457 with A020920, also convolution of A002697 with A038846, also convolution of A002802 with A020918, also convolution of A038845 with A038845. - Rui Duarte, Oct 08 2011

Links

Crossrefs

Cf. A038231.

Programs

  • GAP
    List([0..30], n-> 4^n*Binomial(n+5,5)); # G. C. Greubel, Jul 20 2019
  • Magma
    [4^n*Binomial(n+5, 5): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011
  • Maple
    seq(seq(binomial(i+5, j)*4^i, j =i), i=0..30); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+5,5)*4^n,n=0..30); # Zerinvary Lajos, Jun 16 2008
  • Mathematica
    CoefficientList[Series[1/(1-4x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {24,-240,1280,-3840,6144,-4096}, {1,24,336,3584,32256, 258048}, 30] (* Harvey P. Dale, Mar 24 2018 *)
  • PARI
    Vec(1/(1-4*x)^6 + O(x^30)) \\ Michel Marcus, Aug 21 2015
  • Sage
    [lucas_number2(n, 4, 0)*binomial(n,5)/2^10 for n in range(5, 35)] # Zerinvary Lajos, Mar 11 2009

Formula

a(n) = binomial(n+5, 5)*4^n.
G.f.: 1/(1-4*x)^6.
a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10+i_11+i_12 = n} f(i_1)* f(i_2)*f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10) *f(i_11)*f(i_12), with f(k)=A000984(k). - Rui Duarte, Oct 08 2011
E.g.f.: (15 + 120*x + 240*x^2 + 160*x^3 + 32*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 1620*log(4/3) - 465.
Sum_{n>=0} (-1)^n/a(n) = 12500*log(5/4) - 8365/3. (End)
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