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 12 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)

A364408 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^4).

Original entry on oeis.org

1, 2, -8, 72, -768, 9072, -114240, 1502976, -20414208, 284083968, -4029438976, 58040074752, -846682968064, 12483389708288, -185725854932992, 2784798982701056, -42039464045854720, 638415031298588672, -9746180768647217152, 149486708349609050112
Offset: 0

Views

Author

Seiichi Manyama, Jul 23 2023

Keywords

Crossrefs

Cf. A364393, A364407, A364409.
Cf. A349311.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A349311.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n+4*k-2,n-1) for n > 0.

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.

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

A366327 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^3).

Original entry on oeis.org

1, 2, -5, 33, -260, 2263, -20979, 203124, -2030121, 20786694, -216928144, 2298911699, -24673591005, 267644087524, -2929602893537, 32317666058508, -358931896710948, 4010200327457883, -45040693394259858, 508253687784232108, -5759468659295939684
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2023

Keywords

Crossrefs

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366326, A366328.
Cf. A006632, A364407.

Programs

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

Formula

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

A366358 G.f. satisfies A(x) = 1/(1 - x) + x/A(x)^3.

Original entry on oeis.org

1, 2, -5, 40, -319, 2908, -28151, 284908, -2977115, 31875709, -347884084, 3855802690, -43283239649, 491083601339, -5622489637406, 64877058557080, -753705528179423, 8808460811302729, -103487549564845199, 1221565052783161764, -14480208437556590345
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Crossrefs

Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366356, A366357, A366359.
Cf. A364407, A366327, A366365.

Programs

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

Formula

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

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 12 results. Next

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