A349361 -id:A349361 - cp's OEIS frontend

A349332 G.f. A(x) satisfies A(x) = 1 + x * A(x)^5 / (1 - x).

1, 1, 6, 46, 406, 3901, 39627, 418592, 4551672, 50610692, 572807157, 6577068383, 76426719408, 897078662538, 10620634999318, 126676885170703, 1520759193166329, 18361269213121164, 222814883564042704, 2716125963857227904, 33244557641365865109

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..897

Cf. A002212, A307678, A349331, A349333, A349334, A349335.

Cf. A002294, A346647 (partial sums), A349361, A364748.

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[Binomial[n - 1, k - 1] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^5, 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(5*k,k) / (4*k+1).
a(n) ~ 3381^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Nov 15 2021
Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = (4405*n^4 - 10346*n^3 + 9575*n^2 - 4354*n + 840)*a(n-1) - 12*(n-2)*(1255*n^3 - 3957*n^2 + 4492*n - 1820)*a(n-2) + 2*(n-3)*(n-2)*(10655*n^2 - 32733*n + 26908)*a(n-3) - 4*(n-4)*(n-3)*(n-2)*(3445*n - 6986)*a(n-4) + 3381*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). - Vaclav Kotesovec, Nov 17 2021

A317133 G.f.: Sum_{n>=0} binomial(4(n+1), n)/(n+1) x^n / (1+x)^(n+1).

Original entry on oeis.org

1, 3, 15, 85, 526, 3438, 23358, 163306, 1167235, 8490513, 62648451, 467769217, 3527692298, 26832220834, 205601792340, 1585604105312, 12297768490441, 95861469636203, 750611119223931, 5901214027721577, 46564408929573723, 368644188180241449, 2927350250765841801, 23310167641788680947, 186089697960587977233, 1489085453187335910243

Offset: 0

Paul D. Hanna, Jul 21 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) M(x) = Sum_{n>=0} binomial(2*(n+1), n)/(n+1) * x^n / (1+x)^(n+1) where M(x) = 1 + M(x) + M(x)^2 is the g.f. of Motzkin numbers (A001006).
(C2) 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m.
(C3) 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 + 3*x + 15*x^2 + 85*x^3 + 526*x^4 + 3438*x^5 + 23358*x^6 + 163306*x^7 + 1167235*x^8 + 8490513*x^9 + 62648451*x^10 + ...
such that
A(x) = 1/(1+x) + 4*x/(1+x)^2 + 22*x^2/(1+x)^3 + 140*x^3/(1+x)^4 + 969*x^4/(1+x)^5 + 7084*x^5/(1+x)^6 + ... + A002293(n+1)*x^n/(1+x)^(n+1) + ...
RELATED SERIES.
Series_Reversion( x*A(x) )  =  x/((1+x)^4 - x)  =  x - 3*x^2 + 3*x^3 + 5*x^4 - 22*x^5 + 27*x^6 + 28*x^7 - 163*x^8 + 235*x^9 + 134*x^10 + ...
which equals the sum:
Sum_{n>=0} binomial(n+1, n)/(n+1) * x^(n+1)/(1+x)^(4*(n+1)).

Cf. A316371, A127897, A317134, A349361, A349362, A349363, A349364.

Rest[CoefficientList[InverseSeries[Series[x/((1 + x)^4 - x), {x, 0, 20}], 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))^(1*(m+1)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

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

G.f. A(x) satisfies:
(1) A(x) = (1 + x*A(x))^4 / (1+x).
(2) A(x) = (1/x) * Series_Reversion( x/((1+x)^4 - x) ).
(3) A(x) = Sum_{n>=0} binomial(4*(n+1), n)/(n+1) * x^n / (1+x)^(n+1).
a(n) ~ 229^(n + 3/2) / (sqrt(Pi) * 2^(7/2) * n^(3/2) * 3^(3*n + 9/2)). - Vaclav Kotesovec, Jul 22 2018
a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+1,k) * binomial(4*n-4*k+4,n-k). - Seiichi Manyama, Mar 23 2024

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

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

A371496 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1+x))^4.

Original entry on oeis.org

1, 4, 22, 156, 1233, 10420, 92120, 841376, 7876616, 75177492, 728784802, 7156081536, 71024862452, 711383912672, 7181295333306, 72989746391780, 746308443708928, 7671359593228624, 79226966456758424, 821691132077059740, 8554576791134761387

Offset: 0

Seiichi Manyama, Mar 25 2024

nonn

Cf. A001006, A371494, A371495.

Cf. A118971, A349361.

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n-1, n-k)*binomial(5*k+3, k)/(k+1));

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,n-k) * binomial(5*k+3,k)/(k+1).

G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349361.

A365225 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 + xA(x)^2).

Original entry on oeis.org

1, 1, 4, 24, 169, 1301, 10605, 89963, 785943, 7023148, 63892489, 589771350, 5509967214, 52001860377, 495048989686, 4748144843341, 45838627944500, 445072967642096, 4343508043479012, 42581707009501604, 419158119684986781, 4141270208611084284

Offset: 0

Seiichi Manyama, Aug 27 2023

nonn

Cf. A002293, A271469, A349361, A364759, A364866, A365226.

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

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

A365226 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 + xA(x)^6).

Original entry on oeis.org

1, 1, 4, 20, 107, 577, 3010, 14429, 56640, 98020, -1297568, -21901213, -232421636, -2081040375, -16862259358, -126674303915, -887771735205, -5768588276072, -33971373570320, -170393703586467, -576946353425125, 1101490168511323, 47657979846612682

Offset: 0

Seiichi Manyama, Aug 27 2023

sign

Cf. A002293, A271469, A349361, A364759, A364866, A365225.

Cf. A365218.

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(6*n-k+1, k)*binomial(n-1, n-k)/(6*n-k+1));

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(6*n-k+1,k) * binomial(n-1,n-k)/(6*n-k+1).

A369157 Expansion of (1/x) * Series_Reversion( x / ((1+x)^5-x^5) ).

Original entry on oeis.org

1, 5, 35, 285, 2530, 23750, 231850, 2329850, 23940475, 250394375, 2656849375, 28529354375, 309445377750, 3385369628750, 37312228370000, 413913023212500, 4617886656665625, 51781448191328125, 583266654383859375, 6596645477096428125, 74881064169289121875

Offset: 0

Seiichi Manyama, Jan 15 2024

nonn

Index entries for reversions of series

Cf. A107264, A369156.

Cf. A349361, A369128.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^5-x^5))/x)

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

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

A371544 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1+x))^5.

Original entry on oeis.org

1, 5, 30, 220, 1775, 15206, 135745, 1248900, 11758240, 112736305, 1096960024, 10804727805, 107520029780, 1079346767060, 10917110317185, 111149886462926, 1138205538056395, 11715403351807780, 121137702435412040, 1257720947476195045, 13106870738511517659

Offset: 0

Seiichi Manyama, Mar 26 2024

nonn

Cf. A349361, A371496.

Cf. A371520.

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

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

G.f.: A(x) = B(x)^5 where B(x) is the g.f. of A349361.

cp's OEIS Frontend

A349332 G.f. A(x) satisfies A(x) = 1 + x * A(x)^5 / (1 - x).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A317133 G.f.: Sum_{n>=0} binomial(4*(n+1), n)/(n+1) * x^n / (1+x)^(n+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A371496 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1+x))^4.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A365225 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 + x*A(x)^2).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A365226 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 + x*A(x)^6).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A369157 Expansion of (1/x) * Series_Reversion( x / ((1+x)^5-x^5) ).

A317133 G.f.: Sum_{n>=0} binomial(4(n+1), n)/(n+1) x^n / (1+x)^(n+1).

A365225 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 + xA(x)^2).

A365226 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 + xA(x)^6).