A381831 -id:A381831 - cp's OEIS frontend

A381818 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) / C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

1, 2, 12, 97, 903, 9129, 97419, 1080058, 12319200, 143630575, 1704099034, 20507897766, 249734145622, 3071587654688, 38102046141882, 476138815310364, 5988435287060671, 75745116484532586, 962898676577135634, 12295850972794555196, 157649023155654522723, 2028662477759375282902

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A381817, A381819, A381820.
Cf. A364592, A381830, A381831.
Cf. A000108, A381772.

my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*2*x/(1-sqrt(1-4*x)))^2)/x)^(1/2))

G.f. A(x) satisfies A(x) = C(x*A(x)^2) / (1 - x*A(x)^2).

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

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

Seiichi Manyama, Mar 08 2025

nonn

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

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

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

A381830 G.f. A(x) satisfies A(x) = C(xA(x)^2) / (1 - xA(x)), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 10, 69, 558, 4946, 46506, 455587, 4599494, 47517909, 499933964, 5337957532, 57694565830, 630010984557, 6939976239376, 77027050722166, 860564349616694, 9670164031087137, 109221767288604000, 1239281689627682221, 14119315749935075540, 161460732437631678114

Offset: 0

Seiichi Manyama, Mar 08 2025

nonn

Cf. A364592, A381818, A381831.

Cf. A000108, A381778.

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

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

cp's OEIS Frontend

A381818 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) / C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

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

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Formula

A381830 G.f. A(x) satisfies A(x) = C(xA(x)^2) / (1 - xA(x)), where C(x) is the g.f. of A000108.

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cp's OEIS Frontend

A381818 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) / C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A381830 G.f. A(x) satisfies A(x) = C(x*A(x)^2) / (1 - x*A(x)), where C(x) is the g.f. of A000108.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A381830 G.f. A(x) satisfies A(x) = C(xA(x)^2) / (1 - xA(x)), where C(x) is the g.f. of A000108.