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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A033296 Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis), where each step is (2,1),(1,2) or (1,-1) and start with (1,2).

Original entry on oeis.org

1, 1, 6, 42, 326, 2706, 23526, 211546, 1951494, 18366882, 175674054, 1702686090, 16686795846, 165079509042, 1646340228006, 16534463822010, 167081444125702, 1697551974416706, 17330661859937670, 177699201786231530
Offset: 0

Views

Author

Emeric Deutsch

Keywords

Examples

			G.f. A(x) = 1 + x + 6*x^2 + 42*x^3 + 326*x^4 + 2706*x^5 + 23526*x^6 + 211546*x^7 + 1951494*x^8 + 18366882*x^9 + 175674054*x^10 + ...

Crossrefs

Cf. A027307, A032349, A363308.

Programs

  • PARI
    /* G.f. A(x) = (1/x)*Series_Reversion( x/C(x*C(x)^3) ) */
{a(n) = my(C = (1 - sqrt(1 - 4*x +x^2*O(x^n)))/(2*x)); polcoeff( (1/x)*serreverse(x/subst(C,x,x*C^3)), n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, May 28 2023

Formula

G.f.: A(x) = 1 + x*D(x)^3, where D(x) is the g.f. of A027307. Also: difference of A027307 and A032349. [Changed formula to include a(0) = 1. - Paul D. Hanna, May 28 2023]
D-finite with recurrence +n*(2*n+1)*a(n) +(-32*n^2+47*n-17)*a(n-1) +2*(55*n^2-223*n+228)*a(n-2) +3*(-4*n^2+33*n-70)*a(n-3) -(2*n-7)*(n-5)*a(n-4)=0. - R. J. Mathar, Jul 24 2022
From Paul D. Hanna, May 28 2023: (Start)
G.f. A(x) = (1/x) * Series_Reversion( x / C(x*C(x)^3) ), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
G.f. A(x) = B(x*A(x)) where B(x) = A(x/B(x)) = C(x*C(x)^3) is the g.f. of A363308, and C(x) is the g.f. of the Catalan numbers (A000108). (End)

A108450 Number of pyramids in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis).

Original entry on oeis.org

2, 10, 58, 402, 3122, 26010, 227050, 2049186, 18964194, 178976426, 1715905050, 16665027378, 163611970066, 1621103006010, 16189480081354, 162791835045698, 1646810150270914, 16748008972020554, 171135004105459194
Offset: 1

Views

Author

Emeric Deutsch, Jun 11 2005

Keywords

Comments

A108450(n)=sum(k*A108445(k),k=1..n) (for example, A108450(3)=1*18+2*8+3*8=58). A108450(n)=2*A108453(n). A108450 =2*partial sums of A032349.

Examples

			a(2)=10 because in the A027307(2)=10 paths we have altogether 10 pyramids (shown between parentheses): (ud)(ud), (ud)(Udd), (uudd), uUddd, (Udd)(ud), (Udd)(Udd), Ududd, UdUddd, Uuddd, (UUdddd).

Links

Crossrefs

Cf. A027307, A108445, A108453, A032349.

Programs

  • Maple
    A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=2*z*A^2/(1-z): gser:=series(g,z=0,25): seq(coeff(gser,z^n),n=1..22);
  • Mathematica
    Table[2 Sum[Sum[Binomial[2 k + 2, k - i] Binomial[2 k + i + 1, 2 k + 1], {i, 0, k}]/(k + 1), {k, 0, n - 1}], {n, 19}] (* Michael De Vlieger, Feb 29 2016 *)
  • Maxima
    a(n):=2*sum(sum(binomial(2*k+2,k-i)*binomial(2*k+i+1,2*k+1),i,0,k)/(k+1),k,0,n-1);
/* Vladimir Kruchinin, Feb 29 2016 */
  • PARI
    {a(n)=local(y=2*x); for(i=1, n, y=(2*x*(2+y-x*y)^2)/((1-x)*(2-y+x*y)^2) + (O(x^n))^3); polcoeff(y, n)}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2014

Formula

G.f.: 2*z*A^2/(1-z), where A=1+z*A^2+z*A^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
G.f. y(x) satisfies: y = (2*x*(2+y-x*y)^2)/((1-x)*(2-y+x*y)^2). - Vaclav Kotesovec, Mar 17 2014
a(n) ~ (3*sqrt(5)-1) * ((11+5*sqrt(5))/2)^n /(11*5^(1/4)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 17 2014
a(n) = 2*Sum_{k=0..n-1}(Sum_{i=0..k}(binomial(2*k+2,k-i)* binomial(2*k+i+1,2*k+1))/(k+1)). - Vladimir Kruchinin, Feb 29 2016
D-finite with recurrence n*(2*n-1)*a(n) +6*-(n-1)*(5*n-6)*a(n-1) +4*(23*n^2-97*n+111)*a(n-2) +2*(-29*n^2+142*n-174)*a(n-3) -3*(2*n-5)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

A108453 Number of pyramids of the first kind in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) (a pyramid of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).

Original entry on oeis.org

1, 5, 29, 201, 1561, 13005, 113525, 1024593, 9482097, 89488213, 857952525, 8332513689, 81805985033, 810551503005, 8094740040677, 81395917522849, 823405075135457, 8374004486010277, 85567502052729597, 878066090712156521
Offset: 1

Views

Author

Emeric Deutsch, Jun 11 2005

Keywords

Comments

A108453(n)=sum(k*A108451(k),k=1..n) (for example, A108453(3)=1*16+2*5+3*1=29). A108453(n)=(1/2)*A108450(n). A108453 = partial sums of A032349.

Examples

			a(2)=5 because in the A027307(2)=10 paths we have altogether 5 pyramids of the first kind (shown between parentheses): (ud)(ud), (ud)Udd, (uudd), uUddd, Udd(ud), UddUdd, Ududd, UdUddd, Uuddd, UUdddd.

Links

Crossrefs

Cf. A027307, A108450, A108451, A032349.

Programs

  • Maple
    A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=z*A^2/(1-z): gser:=series(g,z=0,25): seq(coeff(gser,z^n),n=1..22);
  • PARI
    {a(n)=local(y=x); for(i=1, n, y=x*(1+y-x*y)^2/((1-x)*(1-y+x*y)^2) + (O(x^n))^3); polcoeff(y, n)}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2014

Formula

G.f.=zA^2/(1-z), where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
G.f. y(x) satisfies: x*(1+y-x*y)^2 = (1-x)*y*(1-y+x*y)^2. - Vaclav Kotesovec, Mar 17 2014
a(n) ~ sqrt(23*sqrt(5)-15) * (11+5*sqrt(5))^n / (11* sqrt(5*Pi) * n^(3/2) * 2^(n+1/2)). - Vaclav Kotesovec, Mar 17 2014
D-finite with recurrence n*(2*n-1)*a(n) -6*(n-1)*(5*n-6)*a(n-1) +4*(23*n^2-97*n+111)*a(n-2) +2*(-29*n^2+142*n-174)*a(n-3) -3*(2*n-5)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

A371677 G.f. satisfies A(x) = 1 + x * A(x)^(5/2) * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 48, 772, 14256, 285380, 6023552, 131991940, 2974096544, 68475379204, 1603913377040, 38099316926340, 915619571011024, 22222175033464260, 543894269096547296, 13409307961403740420, 332707806061304185408, 8301493488646359256580
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2024

Keywords

Crossrefs

Cf. A006319, A032349, A371675. A371676, A371678.
Cf. A363006.

Programs

  • PARI
    a(n, r=2, t=5, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

G.f. satisfies A(x) = ( 1 + x * A(x)^(5/2) * (1 + A(x)^(1/2)) )^2.
G.f.: B(x)^2 where B(x) is the g.f. of A363006.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(5*n+k+2,n)/(5*n+k+2).

A379251 G.f. A(x) satisfies A(x) = ( (1 + x*A(x))/(1 - x*A(x)^2) )^2.

Original entry on oeis.org

1, 4, 32, 340, 4144, 54724, 761712, 11004500, 163453472, 2480507524, 38292849280, 599455647828, 9493724671184, 151835354054212, 2448792546337360, 39781755539153748, 650386418008379200, 10692713526634029316, 176669496568313495520, 2931998993134971532116, 48854054306918652620912
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A006319, A032349, A379252.
Cf. A361638.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(2*n+3*k+1, k)*binomial(2*n+2*k+2, n-k)/(n+k+1));

Formula

G.f.: B(x)^2 where B(x) is the g.f. of A361638.
a(n) = Sum_{k=0..n} binomial(2*n+3*k+1,k) * binomial(2*n+2*k+2,n-k)/(n+k+1).

A379252 G.f. A(x) satisfies A(x) = ( (1 + x*A(x))/(1 - x*A(x)^3) )^2.

Original entry on oeis.org

1, 4, 40, 572, 9552, 174004, 3352440, 67171500, 1385457568, 29220437860, 627287390664, 13661411796508, 301096488681200, 6703186665881876, 150517000234338072, 3404956079399106700, 77526315562007606080, 1775260286963982001860, 40857405217738915499880, 944584396250976659451388
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A006319, A032349, A379251.
Cf. A379253.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(2*n+5*k+1, k)*binomial(2*n+4*k+2, n-k)/(n+2*k+1));

Formula

G.f.: B(x)^2 where B(x) is the g.f. of A379253.
a(n) = Sum_{k=0..n} binomial(2*n+5*k+1,k) * binomial(2*n+4*k+2,n-k)/(n+2*k+1).

A365622 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^5 ).

Original entry on oeis.org

1, 10, 150, 2670, 52250, 1086002, 23533790, 525825830, 12026993010, 280220428890, 6627397194022, 158692955007390, 3839595257256330, 93725694152075010, 2305406918530451950, 57085385625207424342, 1421808255906105290210
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A032349, A365843, A365847.
Cf. A363006.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+k+4, k)*binomial(5*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(5*(n+1),n-k).
G.f.: B^5, where B is the g.f. of A363006.

A365839 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^2 ).

Original entry on oeis.org

1, 5, 38, 345, 3454, 36786, 408848, 4687969, 55048310, 658645110, 8001060132, 98419541226, 1223430822028, 15344868505700, 193952279202660, 2467977904556145, 31589883835911846, 406463726382152142, 5254324418131556900, 68206459568715464110, 888731044203480723076
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A032349, A365840, A365841.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n+k+2, k)*binomial(2*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(2*(n+1),n-k).

A365840 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^2 ).

Original entry on oeis.org

1, 6, 55, 602, 7263, 93192, 1247636, 17230290, 243669007, 3511010950, 51361157967, 760784343128, 11387857096900, 171988619895216, 2617571721008520, 40105744064042626, 618116513218831407, 9576289414539654450, 149053521972041737413
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A032349, A365839, A365841.
Cf. A278745.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(4*n+k+3, k)*binomial(2*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(2*(n+1),n-k).

A365841 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^2 ).

Original entry on oeis.org

1, 7, 75, 959, 13512, 202433, 3164018, 51010415, 842090988, 14163385916, 241843189651, 4181341506009, 73054000725300, 1287786922627590, 22876030462690500, 409093644922627407, 7358978253387945404, 133067774551068558740, 2417375777620571832476
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A032349, A365839, A365840.
Cf. A365766.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+k+4, k)*binomial(2*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(2*(n+1),n-k).
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