A250886 -id:A250886 - cp's OEIS frontend

A276310 G.f. A(x) satisfies: x = A(x)-2A(x)^2-2A(x)^3.

1, 2, 10, 60, 404, 2912, 21984, 171600, 1373680, 11215776, 93039648, 781936896, 6643741440, 56973685760, 492482782208, 4286561051904, 37536888622848, 330471001126400, 2923338431270400, 25970490200202240, 231607762146309120, 2072719382680535040

Offset: 1

Tom Richardson, Aug 29 2016

			G.f.: A(x) = x + 2*x^2 + 10*x^3 + 60*x^4 + 404*x^5 + 2912*x^6 + 21984*x^7 +...
Related expansions.
A(x)^2 = x^2 + 4*x^3 + 24*x^4 + 160*x^5 + 1148*x^6 + 8640*x^7 + 67296*x^8 +...
A(x)^3 = x^3 + 6*x^4 + 42*x^5 + 308*x^6 + 2352*x^7 + 18504*x^8 +...
where x = A(x) - 2*A(x)^2 - 2*A(x)^3.

Michael De Vlieger, Table of n, a(n) for n = 1..1023
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The three 'R's and the Riordan dual, arXiv:1609.01193 [math.CO], 2016.

Cf. A250886, A276314, A276315, A276316.

Rest[CoefficientList[InverseSeries[Series[x - 2*x^2 - 2*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)

{a(n)=polcoeff(serreverse(x - 2*x^2 - 2*x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

G.f.: Series_Reversion(x - 2*x^2 - 2*x^3).
Conjecture: 3*n*(n-1)*a(n) -13*(n-1)*(2*n-3)*a(n-1) -3*(3*n-5)*(3*n-7)*a(n-2)=0. - R. J. Mathar, Sep 17 2016
a(n) ~ (13 + 5*sqrt(10))^(n - 1/2) / (2^(5/4) * 5^(1/4) * sqrt(Pi) * n^(3/2) * 3^(n - 1/2)). - Vaclav Kotesovec, Aug 22 2017

A276314 G.f. A(x) satisfies: x = A(x)-A(x)^2-3*A(x)^3.

Original entry on oeis.org

1, 1, 5, 20, 104, 546, 3066, 17655, 104555, 630773, 3867617, 24020932, 150827740, 955808680, 6105327912, 39268000188, 254093573088, 1652984379150, 10804631902350, 70925539707330, 467373389649870, 3090558380977020, 20501504119375500, 136392970090612950

Offset: 1

Tom Richardson, Aug 29 2016

			G.f.: A(x) = x+x^2+5*x^3+20*x^4+104*x^5+546*x^6+3066*x^7+... Related Expansions:
A(x)^2=x^2+2*x^3+11*x^4+50*x^5+273*x^6+1500*x^7+8664*x^8+...
A(x)^3=x^3+3*x^4+18*x^5+91*x^6+522*x^7+2997*x^8+17831*x^9+...

Michael De Vlieger, Table of n, a(n) for n = 1..1181
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The three 'R's and the Riordan dual, arXiv:1609.01193 [math.CO], 2016.

Cf. A250886, A276310, A276315, A276316.

Rest[CoefficientList[InverseSeries[Series[x - x^2 - 3*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)

{a(n)=polcoeff(serreverse(x - x^2 - 3*x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

G.f.: Series_Reversion(x-x^2-3*x^3)
Conjecture: +169*n*(n+2)*(n-1)*a(n) +13*(n-1) *(13*n^2+26*n-220) *a(n-1) +(-7277*n^3+13423*n^2+43814*n-81700) *a(n-2) -27*(3*n-10) *(3*n-8) *(71*n+197)*a(n-3)=0. - R. J. Mathar, Sep 17 2016
a(n) ~ (29 + 20*sqrt(10))^(n - 1/2) / (2^(5/4) * 5^(1/4) * sqrt(Pi) * n^(3/2) * 13^(n - 1/2)). - Vaclav Kotesovec, Aug 22 2017

A276315 G.f. A(x) satisfies: x = A(x)-3A(x)^2-2A(x)^3.

Original entry on oeis.org

1, 3, 20, 165, 1524, 15078, 156264, 1674585, 18404980, 206325834, 2350049208, 27118926354, 316381296840, 3725407768140, 44217602683728, 528470024711841, 6354463541900148, 76818345766932450, 933089010748085400, 11382500895815005110, 139387948563917844120

Offset: 1

Tom Richardson, Aug 29 2016

			G.f.: A(x) = x+3*x^2+20*x^3+165*x^4+1524*x^5+15078*x^6+156264*x^7+...
Related Expansions:
A(x)^2 = x^2+6*x^3+49*x^4+450*x^5+4438*x^6+45900*x^7+491181*x^8+...
A(x)^3 = x^3+9*x^4+87*x^5+882*x^6+9282*x^7+100521*x^8+1113299*x^9+...

Michael De Vlieger, Table of n, a(n) for n = 1..897
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The three 'R's and the Riordan dual, arXiv:1609.01193 [math.CO], 2016.

Cf. A250886, A276310, A276314, A276316.

Rest[CoefficientList[InverseSeries[Series[x - 3*x^2 - 2*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)

{a(n)=polcoeff(serreverse(x - 3*x^2 - 2*x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

G.f.: Series_Reversion(x-3*x^2-2*x^3).

a(n) ~ (6*(18 + 5*sqrt(15))/17)^(n - 1/2) / (2*15^(1/4)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 22 2017

A276316 G.f. A(x) satisfies: x = A(x)-4*A(x)^2+A(x)^3.

Original entry on oeis.org

1, 4, 31, 300, 3251, 37744, 459060, 5773548, 74474455, 979872036, 13099102575, 177414673488, 2429310288468, 33574008073120, 467717206216760, 6560977611629676, 92595131510426943, 1313820730347196300, 18730821529411507725, 268185082351558093260

Offset: 1

Tom Richardson, Aug 29 2016

			G.f.: A(x) = x+4*x^2+31*x^3+300*x^4+3251*x^5+37744*x^6+459060*x^7+...
Related Expansions:
A(x)^2 = x^2+8*x^3+78*x^4+848*x^5+9863*x^6+120096*x^7+1511634*x^8+...
A(x)^3 = x^3+12*x^4+141*x^5+1708*x^6+21324*x^7+272988*x^8+3566761*x^9+...

Robert Israel, Table of n, a(n) for n = 1..844
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The three 'R's and the Riordan dual, arXiv:1609.01193 [math.CO], 2016.

Cf. A250886, A276310, A276314, A276315.

S:= series(RootOf(x-4*x^2+x^3-t,x),t,100):
seq(coeff(S,t,j),j=1..100); # Robert Israel, Sep 02 2016

Rest[CoefficientList[InverseSeries[Series[x - 4*x^2 + x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)

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

G.f.: Series_Reversion(x-4*x^2+x^3).
From Robert Israel, Sep 02 2016: (Start)
G.f. g(x) satisfies the differential equation
(12-184*t-27*t^2)*g''(t) - (92+27*t)*g'(t) + 3*g(t) = 4.
(-27*n^2+3)*a(n)+(-184*n^2-276*n-92)*a(n+1)+(12*n^2+36*n+24)*a(n+2) = 0
for n >= 1. (End)
a(n) ~ (46 + 13*sqrt(13))^(n - 1/2) / (13^(1/4) * sqrt(Pi) * n^(3/2) * 2^(n + 1/2) * 3^(n - 1/2)). - Vaclav Kotesovec, Aug 22 2017

A371395 Triangle read by rows: T(n, k) = binomial(n + k, k) * binomial(2*n - k, n - k) / (n + 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 5, 10, 10, 5, 14, 35, 45, 35, 14, 42, 126, 196, 196, 126, 42, 132, 462, 840, 1008, 840, 462, 132, 429, 1716, 3564, 4950, 4950, 3564, 1716, 429, 1430, 6435, 15015, 23595, 27225, 23595, 15015, 6435, 1430

Offset: 0

F. Chapoton, Mar 21 2024

The terms can be seen as graded dimensions of a non-symmetric operad. The Koszul dual operad has Hilbert series x*(1 + x)*(1 + tx). So the current table has as Hilbert series the reverse of x*(1-x)*(1-t*x) w.r.t to x (see Sage below).

The triangle is symmetric under the exchange of k with n - k.

			Triangle begins:
  [0] [ 1],
  [1] [ 1,   1],
  [2] [ 2,   3,   2],
  [3] [ 5,  10,  10,   5],
  [4] [14,  35,  45,  35,  14],
  [5] [42, 126, 196, 196, 126, 42].

Column 0 and main diagonal are A000108.
Column 1 and subdiagonal are A001700.
Row sums are A006013.
The even bisection of the alternating row sums is A001764.
The central terms are A188681.
Cf. A085614, A250886, A371400.

T := (n, k) -> binomial(n + k, k)*binomial(2*n - k, n)/(n + 1):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);  # Peter Luschny, Mar 21 2024

T[n_, k_] := (Hypergeometric2F1[-n, -k, 1, 1] Hypergeometric2F1[-n, k - n, 1, 1]) /(n + 1); Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten
(* Peter Luschny, Mar 21 2024 *)

def Trow(n):
    return [binomial(n+k, k) * binomial(2*n-k, n-k) / (n+1) for k in range(n+1)]

# As the reverse of x*(1-x)*(1-t*x) w.r.t variable x.
t = polygen(QQ, 't')
x = LazyPowerSeriesRing(t.parent(), 'x').0
gf = x*(1-x)*(1-t*x)
coeffs = gf.revert() / x
for n in range(6):
    print(coeffs[n].list())

From Peter Luschny, Mar 21 2024: (Start)
T(n, k) = hypergeom([-n, -k], [1], 1)*hypergeom([-n, k - n], [1], 1)/(n + 1).
2^n*Sum_{k=0..n} T(n, k)*(1/2)^k = A085614(n + 1).
2^n*Sum_{k=0..n} T(n, k)*(-1/2)^k = A250886(n + 1). (End)

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A276310 G.f. A(x) satisfies: x = A(x)-2*A(x)^2-2*A(x)^3.

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A276314 G.f. A(x) satisfies: x = A(x)-A(x)^2-3*A(x)^3.

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A276315 G.f. A(x) satisfies: x = A(x)-3*A(x)^2-2*A(x)^3.

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A276316 G.f. A(x) satisfies: x = A(x)-4*A(x)^2+A(x)^3.

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A371395 Triangle read by rows: T(n, k) = binomial(n + k, k) * binomial(2*n - k, n - k) / (n + 1).

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A276310 G.f. A(x) satisfies: x = A(x)-2A(x)^2-2A(x)^3.

A276315 G.f. A(x) satisfies: x = A(x)-3A(x)^2-2A(x)^3.