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 1-10 of 11 results. Next

A112478 Expansion of (1 + x + sqrt(1 + 6*x + x^2))/2.

Original entry on oeis.org

1, 2, -2, 6, -22, 90, -394, 1806, -8558, 41586, -206098, 1037718, -5293446, 27297738, -142078746, 745387038, -3937603038, 20927156706, -111818026018, 600318853926, -3236724317174, 17518619320890, -95149655201962, 518431875418926, -2832923350929742, 15521467648875090
Offset: 0

Views

Author

Paul Barry, Sep 07 2005

Keywords

Comments

This is the A-sequence for the Delannoy triangle A008288. See the W. Lang link under A006232 for Sheffer a- and z-sequences where also Riordan A- and Z-sequences are explained. O.g.f. A(y) = y/Finv(y) = 2*y/(-(1 + y) + sqrt(y^2 + 6*y + 1)) = ((1 + y) + sqrt(1 + 6*y + y^2))/2 with Finv the inverse function of F(x) = x*(1 + x)/(1 - x). The o.g.f. of the Z-sequence is 1.

Examples

			G.f. = 1 + 2*x - 2*x^2 + 6*x^3 - 22*x^4 + 90*x^5 - 394*x^6 + 1806*x^7 + ...

Links

Crossrefs

A minor variation of A006318. See A085403 for yet another version.
Row sums of number triangle A112477.
Cf. A364393, A364407, A364408, A364409, A366266, A366267, A366268.
Cf. A366325.

Programs

  • Mathematica
    CoefficientList[Series[(1+x+Sqrt[1+6*x+x^2])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
  • PARI
    {a(n) = polcoeff((1 + x + sqrt(1 + 6*x + x^2 + x*O(x^n)))/2, n)}; /* Michael Somos, Jul 07 2020 */

Formula

G.f.: (1 + x + sqrt(1 + 6*x + x^2))/2. - Sergei N. Gladkovskii, Jan 04 2012
G.F.: G(0) where G(k)= 1 + x + x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 04 2012
D-finite with recurrence: n*a(n) + 3*(2*n-3)*a(n-1) + (n-3)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
a(n) ~ (-1)^(n+1) * sqrt(3*sqrt(2) - 4) * (3 + 2*sqrt(2))^n / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014
0 = a(n)*(a(n+1) + 15*a(n+2) + 4*a(n+3)) + a(n+1)*(-3*a(n+1) + 34*a(n+3) + 15*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all integer n > 0. - Michael Somos, Jul 07 2020
From Seiichi Manyama, Oct 08 2023: (Start)
G.f. satisfies A(x) = 1 + x + x/A(x).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(n+k-2,n-k)/(2*k-1). (End)

A366266 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^3.

Original entry on oeis.org

1, 2, 6, 30, 170, 1050, 6846, 46374, 323154, 2301618, 16680246, 122607342, 911868282, 6849381194, 51885977838, 395941193718, 3040818657954, 23485437201762, 182297207394150, 1421357996034750, 11126867651367498, 87421958424703098, 689130671539597854
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2023

Keywords

Crossrefs

Cf. A025227, A366267, A366268.
Cf. A366221, A366364.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(2*k+1,n-k) * binomial(3*k,k)/(2*k+1).
a(n) = A366221(n) + A366221(n-1).
G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366364.

A366267 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^4.

Original entry on oeis.org

1, 2, 8, 56, 448, 3920, 36288, 349440, 3464448, 35125760, 362522624, 3795914240, 40224968704, 430579701760, 4648899846144, 50568103690240, 553632271155200, 6096025799852032, 67464070696927232, 750003531943903232, 8371814935842258944
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2023

Keywords

Crossrefs

Cf. A025227, A366266, A366268.
Cf. A366272, A366365.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = 1 + x + x*A[x]^4 + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(3*k+1, n-k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(3*k+1,n-k) * binomial(4*k,k)/(3*k+1).
a(n) = A366272(n) + A366272(n-1).
G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366365.

A364409 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^5).

Original entry on oeis.org

1, 2, -10, 110, -1430, 20570, -315282, 5047350, -83406510, 1411954610, -24360750810, 426796726334, -7572551327430, 135790011411850, -2457028916693090, 44804882306441990, -822573909558939998, 15191515999168557410, -282038057756813698730
Offset: 0

Views

Author

Seiichi Manyama, Jul 23 2023

Keywords

Links

Crossrefs

Cf. A112478, A364393, A364407, A364408, A366266, A366267, A366268.
Cf. A349312.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A349312.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n+5*k-2,n-1) for n > 0.

A366273 G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^4*A(x)^5.

Original entry on oeis.org

1, 1, 9, 81, 849, 9681, 116601, 1459809, 18809121, 247782369, 3322209001, 45187029809, 621970864241, 8647376531249, 121261376439641, 1713085987837889, 24358211622230081, 348325689458584769, 5006342381846708681, 72279683684984063249, 1047789195353379807121
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2023

Keywords

Crossrefs

Cf. A052709, A366221, A366272.
Cf. A366268.

Programs

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

Formula

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

A366452 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^(5/2).

Original entry on oeis.org

1, 2, 5, 20, 90, 440, 2266, 12110, 66525, 373320, 2130865, 12332512, 72202860, 426861830, 2544727475, 15280236800, 92333523153, 561054410200, 3426075429740, 21013974400920, 129403499560500, 799733464576880, 4958649842375975, 30837325310579350
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366453, A366454, A366455, A366456.
Cf. A259757, A366404.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366404.
a(n) = Sum_{k=0..n} binomial(3*k/2+1,n-k) * binomial(5*k/2,k) / (3*k/2+1).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A259757. - Seiichi Manyama, Apr 04 2024

A366453 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^(7/2).

Original entry on oeis.org

1, 2, 7, 42, 287, 2142, 16898, 138600, 1170037, 10098774, 88712736, 790540296, 7128879940, 64933227996, 596523624144, 5520761026854, 51424824505054, 481741853731110, 4535711525840271, 42897532229559714, 407358615638833341, 3882484733036731500
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366454, A366455, A366456.
Cf. A295537, A366405.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*k/2+1, n-k)*binomial(7*k/2, k)/(5*k/2+1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366405.
a(n) = Sum_{k=0..n} binomial(5*k/2+1,n-k) * binomial(7*k/2,k) / (5*k/2+1).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A295537. - Seiichi Manyama, Apr 04 2024

A366454 G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(3/2).

Original entry on oeis.org

1, 2, -3, 12, -58, 312, -1794, 10794, -67113, 427800, -2780677, 18360504, -122809416, 830379966, -5666465445, 38974338126, -269915089194, 1880576960904, -13172489198859, 92705253700620, -655219698720486, 4648722344211012, -33096948925057703
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366453, A366455, A366456.
Cf. A366400.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366400.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k/2-1,k) * binomial(n+3*k/2-2,n-k) / (5*k/2-1).

A366455 G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(5/2).

Original entry on oeis.org

1, 2, -5, 30, -215, 1710, -14516, 128830, -1180920, 11093830, -106245975, 1033454774, -10181848705, 101394979530, -1018972470275, 10320779179380, -105250097458410, 1079767027094630, -11136159773691830, 115395278542757580, -1200814926210284360
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366453, A366454, A366456.
Cf. A366401.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(7*k/2-1, k)*binomial(n+5*k/2-2, n-k)/(7*k/2-1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366401.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(7*k/2-1,k) * binomial(n+5*k/2-2,n-k) / (7*k/2-1).

A366456 G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(7/2).

Original entry on oeis.org

1, 2, -7, 56, -532, 5600, -62860, 737324, -8929726, 110811344, -1401640814, 18004922936, -234243536436, 3080152906096, -40870739065996, 546563064528906, -7358930622768977, 99672580921800656, -1357142384455626909, 18565841939010374736, -255054402946387767408
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366453, A366454, A366455.
Cf. A366402.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(9*k/2-1, k)*binomial(n+7*k/2-2, n-k)/(9*k/2-1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366402.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(9*k/2-1,k) * binomial(n+7*k/2-2,n-k) / (9*k/2-1).
Showing 1-10 of 11 results. Next

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