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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A366431 G.f. A(x) satisfies A(x) = 1 + x * (A(x) / (1 - x))^(5/2).

Original entry on oeis.org

1, 1, 5, 25, 135, 775, 4651, 28845, 183450, 1190050, 7844230, 52389678, 353770190, 2411324700, 16568343325, 114639216915, 798076174113, 5586035989185, 39287407321075, 277508001643575, 1967816928168265, 14003018984540741, 99965175670335750
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366400.
Cf. A006319, A071724, A213282, A213336, A366432, A366433, A366434, A366435, A366436, A366437.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k/2-1, n-k)*binomial(5*k/2, k)/(3*k/2+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k/2-1,n-k) * binomial(5*k/2,k) / (3*k/2+1).

A371676 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 40, 524, 7824, 126228, 2143544, 37750812, 683194912, 12628104740, 237388091208, 4524456276524, 87228274533040, 1698091537435444, 33332913873239640, 659038408936005692, 13112372856351746112, 262338658739430857796, 5274545338183090647656
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2024

Keywords

Links

Crossrefs

Cf. A006319, A032349, A371675. A371677, A371678.
Cf. A260332.

Programs

  • PARI
    a(n, r=2, t=4, 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)^2 * (1 + A(x)^(1/2)) )^2.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(4*n+k+2,n)/(4*n+k+2).

A371678 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 56, 1068, 23504, 561972, 14183880, 371911132, 10031990560, 276589937892, 7759696110808, 220805824681740, 6357540660485616, 184876232243020564, 5422016433851400552, 160187931368799105468, 4763038761416835095616, 142426926824923660491716
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2024

Keywords

Links

Crossrefs

Cf. A006319, A032349, A371675. A371676, A371677.
Cf. A371700.

Programs

  • PARI
    a(n, r=2, t=6, 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)^3 * (1 + A(x)^(1/2)) )^2.
G.f.: B(x)^2 where B(x) is the g.f. of A371700.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(6*n+k+2,n)/(6*n+k+2).

A247629 Triangular array: T(n,k) = number of paths from (0,0) to (n,k), each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 4, 0, 5, 0, 1, 0, 12, 0, 7, 0, 1, 16, 0, 24, 0, 9, 0, 1, 0, 52, 0, 40, 0, 11, 0, 1, 68, 0, 116, 0, 60, 0, 13, 0, 1, 0, 236, 0, 216, 0, 84, 0, 15, 0, 1, 304, 0, 568, 0, 360, 0, 112, 0, 17, 0, 1, 0, 1108, 0, 1144, 0, 556, 0, 144
Offset: 0

Views

Author

Clark Kimberling, Sep 21 2014

Keywords

Examples

			First 9 rows:
1
0 ... 1
1 ... 0 ... 1
0 ... 3 ... 0 ... 1
4 ... 0 ... 5 ... 0 ... 1
0 ... 12 .. 0 ... 7 ... 0 ...1
16 .. 0 ... 24 .. 0 ... 9 ... 0 ... 1
0 ... 52 .. 0 ... 40 .. 0 ... 11 .. 0 ... 1
68 .. 0 ... 116 . 0 ... 60 .. 0 ... 13 .. 0 ... 1
T(4,2) counts these 5 paths given as vector sums applied to (0,0):
(1,1) + (1,1) + (1,1) + (1,-1)
(1,1) + (1,1) + (2,0)
(1,1) + (1,1) + (1,-1) + (1,1)
(1,1) + (2,0) + (1,1)
(1,1) + (1,-1) + (1,1) + (1,-1)

Links

Crossrefs

Cf. A247623, A247629, A026300, A006319 (1st column of this triangle).

Programs

  • Mathematica
    t[0, 0] = 1; t[1, 1] = 1; t[2, 0] = 1; t[2, 2] = 1; t[n_, k_] := t[n, k] = If[n >= 2 && k >= 1,    t[n - 1, k - 1] + t[n - 1, k + 1] + t[n - 2, k], 0]; t[n_, 0] := t[n, 0] = If[n >= 2, t[n - 2, 0] + t[n - 1, 1], 0]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}]; TableForm[u] (* A247629 array *)
v = Flatten[u] (* A247629 sequence *)
Map[Total, u] (* A247630 *)

A360101 a(n) = Sum_{k=0..n} binomial(n+4*k-1,n-k) * Catalan(k).

Original entry on oeis.org

1, 1, 7, 40, 234, 1432, 9078, 59113, 393125, 2659233, 18240801, 126588424, 887221916, 6271153060, 44652824248, 319990906290, 2306133322704, 16703784324239, 121534039921585, 887845073567240, 6509750423778460, 47888814944642434, 353362258550740732
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Partial sums are A360103.
Cf. A000108, A360057.
Cf. A002212, A006319, A162475, A360100.

Programs

  • Maple
    A360101 := proc(n)
    add(binomial(n+4*k-1,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360101(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    m = 23;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^2/(1 - x)^5 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+4*k-1, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x)^5)))

Formula

G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^5.
G.f.: c(x/(1-x)^5), where c(x) is the g.f. of A000108.
D-finite with recurrence (n+1)*a(n) +(-10*n+7)*a(n-1) +(19*n-56)*a(n-2) +10*(-2*n+9)*a(n-3) +5*(3*n-19)*a(n-4) +(-6*n+49)*a(n-5) +(n-10)*a(n-6)=0. - R. J. Mathar, Mar 12 2023

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).

A083691 Length of list generated by n replacements of k by {-1-|k|, ..., 1+|k|} with increment 2, starting with {0}.

Original entry on oeis.org

1, 2, 6, 20, 76, 296, 1240, 5200, 22960, 100512, 458592, 2064704, 9633472, 44237440, 209780096, 977536256, 4693031680, 22117091840, 107211650560, 509817656320, 2490609167360, 11930278307840, 58656838113280, 282679983493120
Offset: 0

Views

Author

Wouter Meeussen, May 03 2003

Keywords

Comments

G.f. from SuperSeeker (LISTTOALGEQ) checked up to n=11. Same sequence starting with {1}: see A083692. Sum of absolute values of list elements gives A083693. Cross-references cite sequences with similar generation by integer-substitution and length of resulting lists.

Examples

			0, 1 and 2 substitutions produce lengths 1, 2 and 6: {0}; {-1,1}; {-2,0,2, -2,0,2}.

Crossrefs

Cf. A000108, A001003, A006319.
Cf. A083692, A083693.

Programs

  • Mathematica
    Table[Length@Flatten[Nest[ #/.k_Integer:>Table[i, {i, -1-Abs[k], Abs[k]+1, 2}]&, {0}, w]], {w, 0, 10}]
  • PARI
    my(x='x+O('x^33)); Vec( serreverse( (-4*x-5*x^2+x^2*sqrt(1+8*x+8*x^2))/ (2*(-2-6*x-6*x^2-2*x^3)) ) ) \\ Joerg Arndt, Sep 09 2019

Formula

G.f.: 1/x * series_reversion( (-4*x-5*x^2+x^2*sqrt(1+8*x+8*x^2))/ (2*(-2-6*x-6*x^2-2*x^3)) ).

A083692 Length of list generated by n replacements of k by {-1-|k|, .., 1+|k|} with increment 2, starting with {1}.

Original entry on oeis.org

1, 3, 10, 38, 148, 620, 2600, 11480, 50256, 229296, 1032352, 4816736, 22118720, 104890048, 488768128, 2346515840, 11058545920, 53605825280, 254908828160, 1245304583680, 5965139153920, 29328419056640, 141339991746560
Offset: 0

Views

Author

Wouter Meeussen, May 03 2003

Keywords

Comments

Same sequence starting with {0}: see A083691. Sum of absolute values of list elements gives A083693. Cross-references cite sequences with similar generation by integer-substitution and length of resulting lists.

Examples

			0, 1 and 2 substitutions produce lengths 1, 3 and 10:
{1}; {-2,0,2}; {-3,-1,1,3, -1,1, -3,-1,1,3}

Crossrefs

Cf. A000108, A001003, A006319.

Programs

  • Mathematica
    Table[Length@Flatten[Nest[ #/.k_Integer:>Table[i, {i, -1-Abs[k], Abs[k]+1, 2}]&, {1}, w]], {w, 0, 10}]

Formula

Drop first 2 terms from A083691 and divide by 2.
Previous Showing 11-20 of 32 results. Next

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