A106228 -id:A106228 - cp's OEIS frontend

A371655 G.f. satisfies A(x) = 1 + x * A(x) * (1 + A(x))^2.

1, 4, 32, 336, 4032, 52352, 716032, 10161408, 148229120, 2208921600, 33482670080, 514630230016, 8001860567040, 125640146354176, 1989285578473472, 31725578742464512, 509178657425326080, 8217766225008656384, 133287551280741351424, 2171450128344786403328

Offset: 0

Seiichi Manyama, Apr 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..806

Cf. A106228, A107708, A346626.
Cf. A025225, A371658, A371661.
Cf. A100327.

a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 4^(n-k)*binomial(n, k)*binomial(2*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(2*n-k,n-1-2*k) for n > 0.

a(n) = 2^n * A100327(n). - Seiichi Manyama, Dec 26 2024

A378919 G.f. A(x) satisfies A(x) = 1 + xA(x)^6/(1 + xA(x)).

Original entry on oeis.org

1, 1, 5, 39, 355, 3532, 37206, 407861, 4604493, 53169811, 625067441, 7456004083, 90015754691, 1097834790182, 13505674728174, 167395320811562, 2088350145491232, 26203315734195937, 330460721192844017, 4186559092558049570, 53255890990455126082, 679954025388880445771

Offset: 0

Seiichi Manyama, Dec 11 2024

nonn

Cf. A002294, A349362, A364765, A365218, A378892, A378920.

Cf. A106228, A364758, A364759.

a(n, r=1, s=-1, t=6, u=1) = 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))).

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).

A185967 Inverse of Riordan array ((1-x)(1-x^2)(1-x^3)/(1-x^6), x(1-x)(1-x^2)(1-x^3)/(1-x^6)).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 7, 3, 1, 37, 26, 12, 4, 1, 146, 103, 49, 18, 5, 1, 602, 426, 207, 80, 25, 6, 1, 2563, 1818, 897, 359, 120, 33, 7, 1, 11181, 7946, 3966, 1628, 570, 170, 42, 8, 1, 49720, 35389, 17823, 7458, 2701, 852, 231, 52, 9, 1, 224540, 160024, 81177, 34484, 12815, 4212, 1218, 304, 63, 10, 1

Offset: 0

Paul Barry, Feb 07 2011

Riordan array (g(x),xg(x)) where x*g(x) = (x+2)/3 - 2*sqrt(1+x+x^2) * cos(arccos(-(2x^3+3x^2+24x-2) / (2(1+x+x^2)^(3/2)))/3)/3.

			Triangle begins
  1,
  1, 1,
  3, 2, 1,
  10, 7, 3, 1,
  37, 26, 12, 4, 1,
  146, 103, 49, 18, 5, 1,
  602, 426, 207, 80, 25, 6, 1,
  2563, 1818, 897, 359, 120, 33, 7, 1,
  11181, 7946, 3966, 1628, 570, 170, 42, 8, 1,
  49720, 35389, 17823, 7458, 2701, 852, 231, 52, 9, 1,
  224540, 160024, 81177, 34484, 12815, 4212, 1218, 304, 63, 10, 1
Production matrix is
  1, 1,
  2, 1, 1,
  3, 2, 1, 1,
  4, 3, 2, 1, 1,
  5, 4, 3, 2, 1, 1,
  6, 5, 4, 3, 2, 1, 1,
  7, 6, 5, 4, 3, 2, 1, 1,
  8, 7, 6, 5, 4, 3, 2, 1, 1,
  9, 8, 7, 6, 5, 4, 3, 2, 1, 1
  10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1

Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, Bijections on pattern avoiding inversion sequences and related objects, arXiv:2404.04091 [math.CO], 2024. See p. 22.

Inverse of number triangle A185962.

First column is A109081. Row sums are A106228(n+1).

T := (n, k) -> `if`(n=k, 1, (1 + k)*(n - k)*hypergeom([1 + k - n, -n, 1 - k + n], [3/2, 2], 1/4)):
seq(seq(simplify(T(n, k)), k=0..n),n=0..10); # Peter Luschny, Apr 02 2019

T[n_, k_] := (k+1)/(n+1) Sum[Binomial[n+1, i] Binomial[n-k+i-1, n-k-i], {i, 0, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 25 2019, after Vladimir Kruchinin *)

T(n,k):=(k+1)/(n+1)*sum(binomial(n+1,i)*binomial(n-k+i-1,n-k-i),i,0,n-k); /* Vladimir Kruchinin, Apr 02 2019 */

T(n, k) = (k + 1)/(n + 1)*Sum_{i=0..n-k} C(n+1, i)*C(n-k+i-1, n-k-i). - Vladimir Kruchinin, Apr 02 2019

T(n, k) = (1 + k)*(n - k)*hypergeom([1 + k - n, -n, 1 - k + n], [3/2, 2], 1/4) for k < n. - Peter Luschny, Apr 02 2019

A253255 G.f. satisfies: A(x) = (1 - x^3A(x)^3)^2 / (1 - xA(x))^4.

Original entry on oeis.org

1, 4, 26, 202, 1731, 15780, 150117, 1473292, 14807363, 151638550, 1576616125, 16598802248, 176599380271, 1895767748376, 20508188211018, 223348309510194, 2446792909432683, 26944972018189698, 298111489130625351, 3312016395569631402, 36935315970911333184, 413308467174788509668

Offset: 0

Paul D. Hanna, May 31 2015

nonn

Self-convolution of A253256.

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 202*x^3 + 1731*x^4 + 15780*x^5 +...
where A(x) = (1 - x^3*A(x)^3)^2 / (1 - x*A(x))^4.
The logarithm begins:
log(A(x)) = 4*x + 36*x^2/2 + 358*x^3/3 + 3748*x^4/4 + 40404*x^5/5 + 443886*x^6/6 + 4941654*x^7/7 +...+ A168595(n)*x^n/n +...

Cf. A168595, A106228, A064641, A253256.

{a(n) = local(A=1); A = (1/x)*serreverse( x*(1-x)^4/(1-x^3)^2 +x^2*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{A168595(n) = sum(k=0, 2*n, binomial(2*n, k) * polcoeff((1+x+x^2)^n, k) )}
{a(n) = local(A=1); A = exp( sum(k=1,n+1, A168595(k)*x^k/k) +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} A168595(n)*x^n/n ), where A168595(n) = Sum_{k=0..2*n} binomial(n,k)*trinomial(n,k).
(2) A(x) = (1/x)*Series_Reversion( x*(1-x)^4/(1-x^3)^2 ).

A253256 G.f. satisfies: A(x) = (1 - x^3A(x)^6) / (1 - xA(x)^2)^2.

Original entry on oeis.org

1, 2, 11, 79, 647, 5727, 53367, 515802, 5123303, 51977485, 536320688, 5610909773, 59379328267, 634538481389, 6837466955193, 74210071037031, 810527496757335, 8901979424068377, 98253966680382102, 1089260346498608721, 12123804391067414676, 135427509933882292680, 1517725698030921469890

Offset: 0

Paul D. Hanna, May 31 2015

nonn

Self-convolution yields A253255.

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 79*x^3 + 647*x^4 + 5727*x^5 + 53367*x^6 +...
where A(x) = (1 - x^3*A(x)^6) / (1 - x*A(x)^2)^2.
The logarithm begins:
log(A(x)) = 2*x + 18*x^2 + 179*x^3 + 1874*x^4 + 20202*x^5 + 221943*x^6 + 2470827*x^7/7 +...+ A168595(n)/2*x^n/n +...

Cf. A168595, A106228, A064641, A253255.

{a(n) = local(A=1); A = sqrt( (1/x)*serreverse( x*(1-x)^4/(1-x^3)^2 +x^2*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{A168595(n) = sum(k=0, 2*n, binomial(2*n, k) * polcoeff((1+x+x^2)^n, k) )}
{a(n) = local(A=1); A = exp( sum(k=1,n+1, A168595(k)/2 * x^k/k) +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} A168595(n)/2 * x^n/n ), where A168595(n) = Sum_{k=0..2*n} binomial(n,k)*trinomial(n,k).
(2) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-x)^4/(1-x^3)^2 ) ).
(3) A(x) = sqrt( (1-x*A(x) - sqrt(1 - 6*x*A(x) - 3*x^2*A(x)^2)) / (2*x*(1+x*A(x))) ).

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

A365223 G.f. satisfies A(x) = 1 + xA(x)^3 / (1 + xA(x)^4).

Original entry on oeis.org

1, 1, 2, 3, -3, -50, -244, -714, -530, 8522, 63548, 259473, 535647, -1321437, -19094684, -103022071, -322370363, -142186810, 5537336460, 41081448638, 170484444654, 332739198585, -1241023311708, -15677607031084, -83737193010368, -255608722098225, -12706843586158

Offset: 0

Seiichi Manyama, Aug 27 2023

sign

Cf. A000108, A106228, A127897, A364864.

Cf. A365224, A365226.

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

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

A379382 G.f. A(x) satisfies A(x) = sqrt( (1 + 2xA(x))/(1 - 2xA(x)^2) ).

Original entry on oeis.org

1, 2, 8, 48, 336, 2560, 20608, 172416, 1484288, 13062144, 116977664, 1062600704, 9767067648, 90673700864, 848971661312, 8007542571008, 76014137180160, 725681289822208, 6962697126019072, 67105309925048320, 649362348326256640, 6306663216709632000

Offset: 0

Seiichi Manyama, Dec 22 2024

nonn

Cf. A138020, A379383.
Cf. A151374, A379327, A379330.
Cf. A106228.

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

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

a(n) = 2^n * A106228(n).

cp's OEIS Frontend

A371655 G.f. satisfies A(x) = 1 + x * A(x) * (1 + A(x))^2.

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A378919 G.f. A(x) satisfies A(x) = 1 + x*A(x)^6/(1 + x*A(x)).

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A185967 Inverse of Riordan array ((1-x)(1-x^2)(1-x^3)/(1-x^6), x(1-x)(1-x^2)(1-x^3)/(1-x^6)).

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A253255 G.f. satisfies: A(x) = (1 - x^3*A(x)^3)^2 / (1 - x*A(x))^4.

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A253256 G.f. satisfies: A(x) = (1 - x^3*A(x)^6) / (1 - x*A(x)^2)^2.

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

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A365223 G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 + x*A(x)^4).

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A379382 G.f. A(x) satisfies A(x) = sqrt( (1 + 2*x*A(x))/(1 - 2*x*A(x)^2) ).

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A378919 G.f. A(x) satisfies A(x) = 1 + xA(x)^6/(1 + xA(x)).

A253255 G.f. satisfies: A(x) = (1 - x^3A(x)^3)^2 / (1 - xA(x))^4.

A253256 G.f. satisfies: A(x) = (1 - x^3A(x)^6) / (1 - xA(x)^2)^2.

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.

A365223 G.f. satisfies A(x) = 1 + xA(x)^3 / (1 + xA(x)^4).

A379382 G.f. A(x) satisfies A(x) = sqrt( (1 + 2xA(x))/(1 - 2xA(x)^2) ).