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 18 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)

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.

A366268 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^5.

Original entry on oeis.org

1, 2, 10, 90, 930, 10530, 126282, 1576410, 20268930, 266591490, 3569991370, 48509238810, 667157894050, 9269347395490, 129908752970890, 1834347364277530, 26071297610067970, 372683901080814850, 5354668071305293450, 77286026066830771930
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2023

Keywords

Crossrefs

Cf. A025227, A366266, A366267.
Cf. A366273, A366366.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(4*k+1,n-k) * binomial(5*k,k)/(4*k+1).
a(n) = A366273(n) + A366273(n-1).
G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366366.

A366676 G.f. satisfies A(x) = 1 + x^3 + x*A(x)^3.

Original entry on oeis.org

1, 1, 3, 13, 58, 288, 1512, 8250, 46296, 265491, 1548976, 9165156, 54865737, 331694167, 2022232068, 12419023617, 76755164643, 477049187268, 2979758649996, 18695276174079, 117766227611046, 744527923478730, 4722464911515423, 30044091589750350
Offset: 0

Views

Author

Seiichi Manyama, Oct 16 2023

Keywords

Crossrefs

Cf. A176697, A366677.
Cf. A366266, A366555.

Programs

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

Formula

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

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.

A366555 G.f. A(x) satisfies A(x) = 1 + x + x^3*A(x)^3.

Original entry on oeis.org

1, 1, 0, 1, 3, 3, 4, 15, 30, 42, 99, 255, 475, 915, 2232, 4977, 9945, 21945, 51093, 110634, 238005, 542341, 1227390, 2696841, 6035886, 13770402, 31001133, 69485295, 157945293, 359888373, 814699002, 1850816823, 4231092060, 9659302380, 22028018679
Offset: 0

Views

Author

Seiichi Manyama, Oct 13 2023

Keywords

Crossrefs

Cf. A019497, A366266, A366556.
Cf. A366591.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*k+1,n-3*k) * binomial(3*k,k)/(2*k+1).
a(n) = A366591(n) + A366591(n-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

A367640 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^3 / (1 + x)^2.

Original entry on oeis.org

1, 3, 8, 36, 200, 1220, 7896, 53220, 369528, 2624772, 18981864, 139287588, 1034475624, 7761249476, 58735359032, 447827171556, 3436759851672, 26526255859716, 205782644595912, 1603655203428900, 12548225647402248, 98548826076070596, 776552629964300952
Offset: 0

Views

Author

Seiichi Manyama, Nov 25 2023

Keywords

Crossrefs

Cf. A367639, A367641.
Cf. A366221, A366266.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(2*k+2,n-k) * binomial(3*k,k)/(2*k+1).
D-finite with recurrence 2*n*(14*n+71)*(2*n+1)*a(n) +3*(-150*n^3-209*n^2-379*n+228)*a(n-1) +9*(-30*n^3-981*n^2+4297*n-3624)*a(n-2) +27*(n-4)*(22*n^2-491*n+1151)*a(n-3) +81*(n-4)*(n-5)*(6*n-49)*a(n-4)=0. - R. J. Mathar, Dec 04 2023

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).
Showing 1-10 of 18 results. Next

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