A317133 -id:A317133 - cp's OEIS frontend

A349331 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 / (1 - x).

1, 1, 5, 31, 219, 1678, 13570, 114014, 985542, 8708099, 78298727, 714105907, 6590200215, 61427125994, 577456943614, 5468604044500, 52122539760992, 499613409224137, 4813105582181533, 46576519080852235, 452545041339982871, 4413071971740021275, 43177663974461532959

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..985

Cf. A002212, A002293, A307678, A317133, A346646 (partial sums), A349332, A349333, A349334, A349335.

a:= n-> coeff(series(RootOf(1+x*A^4/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..22);  # Alois P. Heinz, Nov 15 2021

nmax = 22; A[] = 0; Do[A[x] = 1 + x A[x]^4/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 22}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^4, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(4*k,k) / (3*k+1).

a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A349362 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^6 / (1 + x).

Original entry on oeis.org

1, 1, 5, 40, 370, 3740, 40006, 445231, 5102165, 59799505, 713496815, 8637432580, 105826926716, 1309793896431, 16351672606365, 205665994855320, 2603696877136060, 33151784577226295, 424258396639960591, 5454120586840761631, 70402732493668027775

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A001006, A002295, A127897, A317133, A346065 (binomial transform), A346666, A349333, A349361, A349363, A349364.

a:= n-> coeff(series(RootOf(1+x*A^6/(1+x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x]^6/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(6*k,k) / (5*k+1).
a(n) = (-1)^(n+1)* F([7/6, 4/3, 3/2, 5/3, 11/6, 1-n], [7/5, 8/5, 9/5, 2, 11/5], 6^6/5^5), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021
a(n) ~ 43531^(n + 1/2) / (72 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 17 2021

A349361 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^5 / (1 + x).

Original entry on oeis.org

1, 1, 4, 26, 194, 1581, 13625, 122120, 1126780, 10631460, 102104845, 994855179, 9809872626, 97710157154, 981636609906, 9935473707279, 101214412755647, 1036991125300748, 10678412226507032, 110459290208905008, 1147261657267290037

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A001006, A002294, A127897, A317133, A345368 (binomial transform), A346665, A349332, A349362, A349363, A349364.

a:= n-> coeff(series(RootOf(1+x*A^5/(1+x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x]^5/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(5*k,k) / (4*k+1).
a(n) = (-1)^(n+1)*F([6/5, 7/5, 8/5, 9/5, 1-n], [3/2, 7/4, 2, 9/4], 5^5/2^8), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021
From Vaclav Kotesovec, Nov 17 2021: (Start)
a(n) ~ 2869^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)).
Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = 3*(615*n^4 - 718*n^3 - 275*n^2 + 618*n - 200)*a(n-1) + 4*(n-2)*(2485*n^3 - 6879*n^2 + 6524*n - 2040)*a(n-2) + 2*(n-3)*(n-2)*(8095*n^2 - 23517*n + 18092)*a(n-3) + 12*(n-4)*(n-3)*(n-2)*(935*n - 1838)*a(n-4) + 2869*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). (End)

A349364 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^8 / (1 + x).

Original entry on oeis.org

1, 1, 7, 77, 987, 13839, 205513, 3176747, 50578445, 823779286, 13660621282, 229865812134, 3915003083306, 67361559577578, 1169138502393414, 20444573270374050, 359858503314494318, 6370677542063831319, 113359050598950194801, 2026309136822686950087

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(m*k,k) / ((m-1)*k+1) ~ (m-1)^(m/2 - 2) * (m^m/(m-1)^(m-1) - 1)^(n + 1/2) / (sqrt(2*Pi) * m^((m-1)/2) * n^(3/2)). - Vaclav Kotesovec, Nov 17 2021

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A001006, A007556, A127897, A317133, A346668, A346672 (binomial transform), A349335, A349361, A349362, A349363.

a:= n-> coeff(series(RootOf(1+x*A^8/(1+x)-A, A), x, n+1), x, n):
seq(a(n), n=0..19);  # Alois P. Heinz, Nov 15 2021

nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^8/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(8*k,k) / (7*k+1).
a(n) = (-1)^(n+1)*F([9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8, 1-n], [9/7, 10/7, 11/7, 12/7, 13/7, 2, 15/7], 8^8/7^7), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021
a(n) ~ 15953673^(n + 1/2) / (2048 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 17 2021

A349363 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^7 / (1 + x).

Original entry on oeis.org

1, 1, 6, 57, 629, 7589, 96942, 1288729, 17643920, 247089010, 3522891561, 50964747400, 746241617226, 11038241689188, 164696773030055, 2475832560808858, 37462189433509758, 570112127356828846, 8720472842436039280, 133997057207982607092, 2067402314984991892461

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A001006, A002296, A127897, A317133, A346667, A346671 (binomial transform), A349334, A349361, A349362, A349364.

a:= n-> coeff(series(RootOf(1+x*A^7/(1+x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x]^7/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(7*k,k) / (6*k+1).
a(n) = (-1)^(n+1)* F([8/7, 9/7, 10/7, 11/7, 12/7, 13/7, 1-n], [4/3, 3/2, 5/3, 11/6, 2, 13/6], 7^7/6^6), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021
a(n) ~ 776887^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Nov 17 2021

A317134 G.f.: Sum_{n>=0} binomial(4(n+1), n)/(n+1) x^n / (1+x)^(2*(n+1)).

Original entry on oeis.org

1, 2, 9, 44, 236, 1336, 7862, 47608, 294720, 1856748, 11865684, 76731572, 501176237, 3301501694, 21909634763, 146337236580, 982962605577, 6635968279354, 45001173711683, 306406562117884, 2093909763907401, 14356806252396614, 98735015302171955, 680906548260420320, 4707709357806093085, 32625093782844333722, 226588405850230665429, 1576882804780751603092

Offset: 0

Paul D. Hanna, Jul 22 2018

nonn

Note that: binomial(4*(n+1), n)/(n+1) = A002293(n+1) for n >= 0, where F(x) = Sum_{n>=0} A002293(n)*x^n satisfies F(x) = 1 + x*F(x)^4.
Compare the g.f. to:
(C1) 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m.
(C2) If S(x,p,q) = Sum_{n>=0} binomial(p*(n+1),n)/(n+1) * x^n/(1+x)^(q*(n+1)), then Series_Reversion ( x*S(x,p,q) ) = x*S(x,q,p) holds for fixed p and q.

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 44*x^3 + 236*x^4 + 1336*x^5 + 7862*x^6 + 47608*x^7 + 294720*x^8 + 1856748*x^9 + 11865684*x^10 + ...
such that
A(x) = 1/(1+x)^2 + 4*x/(1+x)^4 + 22*x^2/(1+x)^6 + 140*x^3/(1+x)^8 + 969*x^4/(1+x)^10 + 7084*x^5/(1+x)^12 + ... + A002293(n+1)*x^n/(1+x)^(2*(n+1)) + ...
RELATED SERIES.
Series_Reversion( x*A(x) )  =  4*x/((1+x)^2 + sqrt( (1+x)^4 - 4*x ))^2  =  x - 2*x^2 - x^3 + 6*x^4 + 3*x^5 - 20*x^6 - 18*x^7 + 74*x^8 + 111*x^9 - 278*x^10 - 657*x^11 + 980*x^12 + 3739*x^13 + ...
which equals the sum:
Sum_{n>=0} binomial(2*(n+1), n)/(n+1) * x^(n+1)/(1+x)^(4*(n+1)).
The square-root of the g.f. is an integer series:
sqrt(A(x)) = 1 + x + 4*x^2 + 18*x^3 + 92*x^4 + 504*x^5 + 2897*x^6 + 17235*x^7 + 105233*x^8 + 655687*x^9 + 4152461*x^10 + ... + A317135(n)*x^n + ...
which equals the sum:
Sum_{n>=0} binomial(4*n+2, n)/(2*n+1) * x^(n+1)/(1+x)^(2*n+1).

Cf. A316371, A127897, A316987, A317133.

Rest[CoefficientList[InverseSeries[Series[4*x/((1 + x)^2 + Sqrt[(1 + x)^4 - 4*x])^2, {x, 0, 30}], x], x]](* Vaclav Kotesovec, Jul 22 2018 *)

{a(n) = my(A = sum(m=0, n, binomial(4*(m+1), m)/(m+1) * x^m / (1+x +x*O(x^n))^(2*(m+1)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

{a(n) = my(A = (1/x) * serreverse( 4*x/((1+x)^2 + sqrt( (1+x)^4 - 4*x  + x*O(x^n)))^2 )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) A(x) = ( (1 + x*A(x))^2 + sqrt( (1 + x*A(x))^4 - 4*x*A(x) ) )^2 / 4.
(2) A(x) = (1/x) * Series_Reversion( 4*x/((1+x)^2 + sqrt( (1+x)^4 - 4*x ))^2 ).
(3) A(x) = Sum_{n>=0} binomial(4*(n+1), n)/(n+1) * x^n / (1+x)^(2*(n+1)).
a(n) ~ 37^(1/4) * (101 + 16*sqrt(37))^(n+1) / (2*sqrt(Pi) * n^(3/2) * 3^(3*n + 9/2)). - Vaclav Kotesovec, Jul 22 2018

A365224 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 + xA(x)^5).

Original entry on oeis.org

1, 1, 3, 10, 30, 56, -167, -2813, -21515, -126135, -601812, -2179039, -3455504, 32238155, 430944400, 3334419890, 20083350422, 97094186751, 338485665435, 274332822425, -8491831747320, -97735154210032, -732963337489636, -4341176221239330

Offset: 0

Seiichi Manyama, Aug 27 2023

sign

Cf. A001764, A219537, A317133, A364758, A364865.

Cf. A365223, A365226.

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

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

A369156 Expansion of (1/x) * Series_Reversion( x / ((1+x)^4-x^4) ).

Original entry on oeis.org

1, 4, 22, 140, 968, 7064, 53544, 417456, 3326304, 26967040, 221733568, 1844667136, 15498804480, 131325820032, 1120928667264, 9628975973120, 83181462291968, 722175844640768, 6297942966129664, 55143987250677760, 484589284705202176, 4272491458636754944

Offset: 0

Seiichi Manyama, Jan 15 2024

nonn

Index entries for reversions of series

Cf. A107264, A369157.

Cf. A317133, A369126.

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

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

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

A377458 G.f. A(x) satisfies A(x) = 1 + x/A(x)^2 * (1 - A(x) + A(x)^4).

Original entry on oeis.org

1, 1, 1, 4, 7, 29, 61, 256, 596, 2507, 6247, 26197, 68652, 286232, 780508, 3231060, 9102590, 37392935, 108279767, 441342883, 1308552478, 5292781266, 16018989626, 64315663716, 198213843417, 790252270626, 2474924176566, 9802205324516, 31142246753638

Offset: 0

Seiichi Manyama, Dec 12 2024

nonn

Cf. A317133, A364759, A377706, A378958, A378920.

Cf. A371891.

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

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

A377706 G.f. A(x) satisfies A(x) = 1 + x/A(x)^3 * (1 - A(x) + A(x)^4).

Original entry on oeis.org

1, 1, 0, 3, -6, 28, -105, 444, -1897, 8338, -37305, 169471, -779537, 3623500, -16993990, 80316081, -382136133, 1828896726, -8798796709, 42528048930, -206413678447, 1005623593109, -4916026689088, 24106987842416, -118551374861525, 584526569727010, -2888995759466360

Offset: 0

Seiichi Manyama, Dec 12 2024

sign

Cf. A317133, A364759, A377458, A378958, A378920.

Cf. A371890.

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

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

cp's OEIS Frontend

A349331 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 / (1 - x).

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A349362 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^6 / (1 + x).

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A349361 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^5 / (1 + x).

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A349364 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^8 / (1 + x).

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A349363 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^7 / (1 + x).

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A317134 G.f.: Sum_{n>=0} binomial(4*(n+1), n)/(n+1) * x^n / (1+x)^(2*(n+1)).

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A365224 G.f. satisfies A(x) = 1 + x*A(x)^4 / (1 + x*A(x)^5).

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A369156 Expansion of (1/x) * Series_Reversion( x / ((1+x)^4-x^4) ).

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A317134 G.f.: Sum_{n>=0} binomial(4(n+1), n)/(n+1) x^n / (1+x)^(2*(n+1)).

A365224 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 + xA(x)^5).