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 1-5 of 5 results.

A368969 Expansion of (1/x) * Series_Reversion( x * (1-x+x^2)^2 ).

Original entry on oeis.org

1, 2, 5, 12, 22, 0, -284, -1938, -9367, -36938, -118105, -260130, 56637, 4890560, 35945616, 186674620, 782890326, 2632462236, 5987222046, -2241224328, -129137211280, -967479390360, -5145272296080, -22060975744080, -75535676951124, -172915138783080
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Links

Crossrefs

Cf. A368973, A368975.
Cf. A368961.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x+x^2)^2)/x)
  • PARI
    a(n, s=2, t=2, u=0) = sum(k=0, n\s, (-1)^k*binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

Formula

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

A368973 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x+x^2)^2 ).

Original entry on oeis.org

1, 3, 13, 65, 351, 1989, 11650, 69903, 427225, 2649229, 16622079, 105310673, 672687322, 4327037010, 28002409452, 182179075689, 1190778886791, 7815755146095, 51491064226095, 340374137775879, 2256891800364421, 15006481967365535, 100037043223408890
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Links

Crossrefs

Cf. A368969, A368975.
Cf. A368965.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x+x^2)^2)/x)
  • PARI
    a(n, s=2, t=2, u=1) = sum(k=0, n\s, (-1)^k*binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

Formula

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

A381828 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^2 ) )^(1/2).

Original entry on oeis.org

1, 2, 10, 65, 480, 3824, 32039, 278256, 2482578, 22617830, 209540672, 1968031520, 18696064179, 179332892186, 1734451272240, 16895744042472, 165621305486976, 1632518433458400, 16170959983623314, 160888256475481560, 1607061512154585046, 16110030923830784248
Offset: 0

Views

Author

Seiichi Manyama, Mar 08 2025

Keywords

Crossrefs

Cf. A188687, A381817, A381829.
Cf. A129442, A381831.
Cf. A000108, A368975.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*(1-x+x^2))^2)/x)^(1/2))

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x)^2), where C(x) is the g.f. of A000108.
a(n) = Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(3*n-2*k,n-k)/(2*n+k+1).
D-finite with recurrence +432*n*(n-1)*(n-2)*(2*n+1)*(2*n-1)*(2*n-3)*(262261060139434887136491*n -880264534325728808928710)*a(n) +24*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(9441398165019655936913676*n^3 -1563359509176097527827297363*n^2 +8122005300033248841454135898*n -10005843136737488906545668303)*a(n-1) -8*(n-2)*(2*n-3)*(26904862014415612504704360259*n^5 -439294650192331167438487778367*n^4 +2462557164881954865201862193560*n^3 -6116391863054255517662202621591*n^2 +6730597164009721987374566778403*n -2508886036978141982914230533400)*a(n-2) +2*(3280856375160701992555505608813*n^7 -60505233834440544774094319915261*n^6 +458650706405377012453301766859297*n^5 -1843996542698657351167896639498197*n^4 +4199211312282774397146042070543498*n^3 -5283107978583820687249123910721062*n^2 +3195330463869279708956264243293272*n -571272270914692694572799416918200)*a(n-3) +3*(-10499174187769013704183946812135*n^7 +189831332911960443054698384732480*n^6 -1395267797131742288585801071743534*n^5 +5221938509132769354051685228032464*n^4 -9839826026184653630837080778918103*n^3 +6229383740555425356174546560814416*n^2 +6216439623275682391743799709941612*n -8390747283534155728971424365124320)*a(n-4) -112*(7*n-31)*(7*n-32) *(2094251874056865218841652*n -5622141652266976856940223)*(7*n-29)*(7*n-26) *(7*n-30)*(7*n-27)*a(n-5)=0. - R. J. Mathar, Mar 10 2025

A369076 Expansion of (1/x) * Series_Reversion( x * (1+x^2/(1-x))^2 ).

Original entry on oeis.org

1, 0, -2, -2, 9, 24, -37, -240, -2, 2126, 2919, -16052, -50663, 86940, 631995, 19094, -6491463, -9595434, 54443985, 181532910, -317331187, -2426618056, -133151895, 26332109928, 40544827703, -230619508548, -793966990358, 1384746844832, 10960715925621, 881359815524
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2024

Keywords

Links

Crossrefs

Cf. A168491, A369077.
Cf. A368969, A368973, A368975.
Cf. A368957.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1+x^2/(1-x))^2)/x)
  • PARI
    a(n, s=2, t=2, u=-2) = sum(k=0, n\s, (-1)^k*binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

Formula

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

A372464 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-x+x^2) )^(2*n).

Original entry on oeis.org

1, 4, 32, 286, 2688, 26004, 256334, 2560352, 25824768, 262447684, 2683152032, 27565067600, 284330359950, 2942808943572, 30546407611136, 317867390671536, 3314979452815360, 34637849797078380, 362544825234198020, 3800439733237986800, 39893311092729794688
Offset: 0

Views

Author

Seiichi Manyama, May 01 2024

Keywords

Crossrefs

Cf. A368975, A372460.

Programs

  • PARI
    a(n, s=2, t=2, u=2) = sum(k=0, n\s, (-1)^k*binomial(t*n+k-1, k)*binomial((t+u+1)*n-(s-1)*k-1, n-s*k));

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n+k-1,k) * binomial(5*n-k-1,n-2*k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x)^2 * (1-x+x^2)^2 ). See A368975.
Showing 1-5 of 5 results.

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