A219537 -id:A219537 - cp's OEIS frontend

A370473 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 - A(x) + A(x)^2 - A(x)^3 + A(x)^4).

1, 1, 4, 25, 185, 1501, 12914, 115723, 1068505, 10094770, 97117624, 948181724, 9370734322, 93562986440, 942385174150, 9563720899515, 97696642766654, 1003789888620166, 10366477185870960, 107548800153957745, 1120374840689934195, 11714707429579539268

Offset: 0

Seiichi Manyama, Mar 31 2024

nonn

Cf. A000108, A219537.
Cf. A370472, A370476.
Cf. A366401.

a(n) = sum(k=0, n, binomial(n, k)*binomial(n+5*k/2+1/2, n)/(2*n+5*k+1));

G.f. A(x) satisfies:
(1) A(x)^2 = 1 + x * A(x)^2 * (1 + A(x)^5).
(2) A(x) = sqrt(B(x)) where B(x) is the g.f. of A366401.
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+5*k/2+1/2,n)/(2*n+5*k+1).

A371658 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x))^2.

Original entry on oeis.org

1, 4, 48, 784, 14784, 302976, 6555648, 147380480, 3408817152, 80592320512, 1938923790336, 47314993324032, 1168315059240960, 29136848453632000, 732857340425011200, 18569095605771632640, 473534596510970019840, 12144227894941523116032

Offset: 0

Seiichi Manyama, Apr 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..694

Cf. A219534, A219537, A371657.
Cf. A025225, A371655, A371661.
Cf. A219538.

a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 4^(n-k)*binomial(n, k)*binomial(3*n-k, n-1-2*k))/n);

a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 4^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-2*k) for n > 0.

a(n) = 2^n * A219538(n). - Seiichi Manyama, Dec 26 2024

A378920 G.f. A(x) satisfies A(x) = 1 + xA(x)^6/(1 + xA(x)^2).

Original entry on oeis.org

1, 1, 5, 38, 339, 3308, 34191, 367844, 4076112, 46204209, 533239820, 6244542391, 74016115926, 886276231388, 10704869669941, 130271156244371, 1595708949486866, 19658780721376791, 243429900033986385, 3028086095940468087, 37821457123957529163, 474145963420441744445

Offset: 0

Seiichi Manyama, Dec 11 2024

nonn

Cf. A002294, A349362, A364765, A365218, A378892, A378919.

Cf. A000108, A219537, A291534, A365225.

a(n, r=1, s=-1, t=6, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

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

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r).

A364740 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 - xA(x)^5).

Original entry on oeis.org

1, 1, 3, 15, 91, 607, 4298, 31720, 241321, 1879097, 14903013, 119965086, 977623639, 8049579047, 66864689674, 559650696185, 4715304229460, 39960204165865, 340395043021399, 2912963919210012, 25031055321749916, 215894227588453950, 1868403327770467149

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Cf. A001003, A001764, A219537, A364739.

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

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

A364833 G.f. satisfies A(x) = 1 + xA(x)^2/(1 - x^3A(x)^3).

Original entry on oeis.org

1, 1, 2, 5, 15, 49, 168, 595, 2160, 7997, 30083, 114660, 441840, 1718531, 6737820, 26600784, 105659970, 421949492, 1693120779, 6823018035, 27602090087, 112053680381, 456343848121, 1863893501065, 7633232165286, 31337360839387, 128944120202510

Offset: 0

Seiichi Manyama, Aug 28 2023

nonn

Cf. A112806, A219537.
Cf. A218251, A364161, A365247.
Cf. A001003, A049124.

A364833 := proc(n)
    add( binomial(n-2*k-1,k)*binomial(2*n-3*k+1,n-3*k)/ (2*n-3*k+1),k=0..floor(n/3)) ;
end proc:
seq(A364833(n),n=0..80); # R. J. Mathar, Aug 29 2023

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

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(2*n-3*k+1,n-3*k)/(2*n-3*k+1).
D-finite with recurrence 31*n*(626109182191*n-1858292669035) *(n-1)*(n+1) *a(n) -n*(n-1) *(244150473843619*n^2 -1454194662255591*n +2175006457069082) *a(n-1) +3*(n-1) *(292927551362415*n^3 -2593205532882651*n^2 +7084566217454162*n -5823331737745632)*a(n-2) +(-843955616916167*n^4 +9932491073296715*n^3 -42016891739306929*n^2 +76184884157722453*n -50166914106142776) *a(n-3) +18*(1509721335071*n^4 -40413442328880*n^3 +330301781039401*n^2 -1078322794857576*n +1231650372542192) *a(n-4) +18*(39673125909769*n^4 -598320530478001*n^3 +3228489073613917*n^2 -7321259523567459*n +5788776339353646) *a(n-5) +27*(n-5) *(3102413205331*n^3 -35996479327373*n^2 +114122791959960*n -64735736097804) *a(n-6) -243*(n-6) *(n-7)*(475638134099*n^2 -2399948859181*n +2877042451214) *a(n-7) -243*(45857481910*n -35520400961) *(n-5) *(n-7) *(n-8)*a(n-8)=0. - R. J. Mathar, Aug 29 2023
G.f.: (1/x) * Series_Reversion( x*(1 - x / (1 - x^3)) ). - Seiichi Manyama, Sep 28 2024

A363573 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).

Original entry on oeis.org

1, 1, 3, 16, 99, 670, 4804, 35855, 275635, 2167577, 17354844, 140994899, 1159398760, 9631155422, 80703507043, 681333999628, 5789823864323, 49484286592503, 425092050147999, 3668385302806058, 31786451503719132, 276447315011186576, 2412336247105063011, 21114946136742383146

Offset: 0

Paul D. Hanna, Aug 14 2023

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 99*x^4 + 670*x^5 + 4804*x^6 + 35855*x^7 + 275635*x^8 + 2167577*x^9 + 17354844*x^10 + ...
such that
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
RELATED TABLE.
The table of coefficients in A(x)^n begins:
n=1: [1, 1,  3,  16,   99,   670,   4804,   35855,   275635, ...];
n=2: [1, 2,  7,  38,  239,  1634,  11798,   88506,   683045, ...];
n=3: [1, 3, 12,  67,  429,  2967,  21594,  162945,  1263183, ...];
n=4: [1, 4, 18, 104,  679,  4756,  34922,  265244,  2066591, ...];
n=5: [1, 5, 25, 150, 1000,  7101,  52645,  402725,  3155125, ...];
n=6: [1, 6, 33, 206, 1404, 10116,  75775,  584148,  4603911, ...];
n=7: [1, 7, 42, 273, 1904, 13930, 105490,  819918,  6503553, ...];
n=8: [1, 8, 52, 352, 2514, 18688, 143152, 1122312,  8962615, ...];
n=9: [1, 9, 63, 444, 3249, 24552, 190326, 1505727, 12110400, ...];
...
from which one can verify the formulas involving powers of A(x).
RELATED SERIES.
Let G(x) = 1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^3) )
where
G(x) = 1 + x + 2*x^2 + 9*x^3 + 42*x^4 + 219*x^5 + 1202*x^6 + 6867*x^7 + 40378*x^8 + 242782*x^9 + 1485836*x^10 + ...
then
A(x) = G(x*A(x)),
and so
A(x) = (1/x) * Series_Reversion( x/G(x) );
thus,
x*A(x) = (A(x) - 1) / (1 + (A(x) - 1)*(A(x)^2 + A(x)^3) )
which is equivalent to
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).

Cf. A219537, A271469, A300048, A161634, A363560.

a:= n-> coeff(series(RootOf(1-A+x*(A-A^3+A^5), A), x, n+1), x, n):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 14 2023

{a(n) = my(A=1+x); for(i=1, n, A = 1 + x*(A - A^3 + A^5) +x*O(x^n) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
(2) A(x)^2 = 1 + x*(A(x) + A(x)^2 - A(x)^3 - A(x)^4 + A(x)^5 + A(x)^6).
(3) A(x)^3 = 1 + x*(A(x) + A(x)^2 - A(x)^4 + A(x)^6 + A(x)^7).
(4) A(x)^4 = 1 + x*(A(x) + A(x)^2 + A(x)^7 + A(x)^8).
(5) A(x)^5 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^8 + A(x)^9).
(6) A(x)^6 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^6 + A(x)^9 + A(x)^10).
(7) A(x)^7 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^10 + A(x)^11).
(8) A(x) = (1/x) * Series_Reversion( x/(1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^3) ) ) ).
(9) A(x) = 1 / A(-x*A(x)^5).

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).

A369478 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^2)^2) ).

Original entry on oeis.org

1, 4, 24, 170, 1320, 10868, 93197, 823484, 7445184, 68545882, 640446224, 6057249180, 57878746750, 557903174040, 5418441862824, 52971933934834, 520869559359424, 5147999004530720, 51113415228327827, 509583784051748692, 5099262428810825568

Offset: 0

Seiichi Manyama, Jan 23 2024

nonn

Index entries for reversions of series

Cf. A219537, A369480.
Cf. A143927, A369477.
Cf. A369441.

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

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

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

A371657 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x) + A(x)^2).

Original entry on oeis.org

1, 3, 27, 333, 4752, 73764, 1209492, 20610693, 361403937, 6478386561, 118181952369, 2186908154748, 40949739595242, 774474351098031, 14772979729013247, 283878381945510621, 5490264493926636912, 106786725176131118523, 2087502569999563971843

Offset: 0

Seiichi Manyama, Apr 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..756

Cf. A107708, A156016.

Cf. A219534, A219537, A371658.

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

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

A333472 a(n) = [x^n] ( c (x/(1 + x)) )^n, where c(x) = (1 - sqrt(1 - 4x))/(2x) is the o.g.f. of the Catalan numbers A000108.

Original entry on oeis.org

1, 1, 3, 13, 59, 276, 1317, 6371, 31131, 153292, 759428, 3780888, 18900389, 94805959, 476945913, 2405454213, 12158471195, 61574325840, 312365992620, 1587052145492, 8074474510884, 41131551386120, 209760563456920, 1070822078321520, 5471643738383781, 27982867986637151

Offset: 0

Peter Bala, Mar 23 2020

Let F(x) = 1 + f(1)*x + f(2)*x^2 + ... be a power series with integer coefficients. The associated sequence u(n) := [x^n] F(x)^n is known to satisfy the Gauss congruences: u(n*p^k) == u(n*p^(k-1)) ( mod p^k ) for any prime p and positive integers n and k. For certain power series F(x), stronger congruences may hold. Examples include F(x) = (1 + x)^2, F(x) = 1/(1 - x) and F(x) = c(x), where c(x) is the o.g.f. of the Catalan numbers A000108. The associated sequences (with some differences of offset) are A000984, A001700 and A025174, respectively.
Here we take F(x) = c(x/(1 + x)) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 21*x^6 + ... (cf. A001006 and A086246) and conjecture that the associated sequence a(n) = [x^n] ( c(x/(1 + x)) )^n satisfies the supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for prime p >= 5 and positive integers n and k. Cf. A333473.
More generally, we conjecture that for any positive integer r and any integer s the sequence a(r,s;n) := [x^(r*n)] ( c(x/(1 + x)) )^(s*n) also satisfies the above congruences.
Note that the sequence b(n) := [x^n] c(x)^n = A025174(n) satisfies the stronger congruences b(n*p^k) == b(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. The sequence d(n) := [x^n] ( (1 + x)*c(x/(1 + x)) )^n = A333093(n) appears to satisfy the same congruences.

			Examples of congruences:
a(11) - a(1) = 3780888 - 1 = (11^2)*31247 == 0 ( mod 11^2 ).
a(3*7) - a(3) = 41131551386120 - 13 = (7^2)*13*23671*2727841 == 0 ( mod 7^2 ).
a(5^2) - a(5) = 27982867986637151 - 276 = (5^4)*13*74687*46113049 == 0 ( mod 5^4 ).

Cf. A000108, A001006, A086246, A106228, A219537, A333093, A333473.

Cat := x -> (1/2)*(1-sqrt(1-4*x))/x:
G := x -> Cat(x/(1+x)):
H := (x,n) -> series(G(x)^n, x, 51):
seq(coeff(H(x, n), x, n), n = 0..25);

Table[SeriesCoefficient[((1 + x - Sqrt[1 - 2*x - 3*x^2]) / (2*x))^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 29 2020 *)

a(n) = [x^n] ( (1 + x - sqrt(1 - 2*x - 3*x^2)) / (2*x) )^n.

a(n) ~ sqrt(((9386 + 1026*sqrt(57))^(1/3) + (9386 - 1026*sqrt(57))^(1/3) - 19)/228) * (((1261 + 57*sqrt(57))^(1/3) + (1261 - 57*sqrt(57))^(1/3) + 10)/6)^n / sqrt(Pi*n). - Vaclav Kotesovec, Mar 29 2020

cp's OEIS Frontend

A370473 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 - A(x) + A(x)^2 - A(x)^3 + A(x)^4).

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A371658 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x))^2.

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A378920 G.f. A(x) satisfies A(x) = 1 + x*A(x)^6/(1 + x*A(x)^2).

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A364740 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x)^5).

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A364833 G.f. satisfies A(x) = 1 + x*A(x)^2/(1 - x^3*A(x)^3).

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A363573 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).

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

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A369478 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^2)^2) ).

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A371657 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x) + A(x)^2).

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A378920 G.f. A(x) satisfies A(x) = 1 + xA(x)^6/(1 + xA(x)^2).

A364740 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 - xA(x)^5).

A364833 G.f. satisfies A(x) = 1 + xA(x)^2/(1 - x^3A(x)^3).

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

A333472 a(n) = [x^n] ( c (x/(1 + x)) )^n, where c(x) = (1 - sqrt(1 - 4x))/(2x) is the o.g.f. of the Catalan numbers A000108.