A063020 -id:A063020 - cp's OEIS frontend

A007858 G.f. is 1 - 1/f(x), where f(x) = 1+x+3x^2+9x^3+32*x^4+... is 1/x times g.f. for A063020.

1, 2, 4, 13, 44, 164, 636, 2559, 10556, 44440, 190112, 824135, 3612244, 15981632, 71277736, 320121747, 1446537564, 6571858168, 30000766128, 137544893940, 633051803120, 2923867281660, 13547594977500, 62955434735505, 293336372858724, 1370149533359784, 6414423856436816

Offset: 1

Martin Klazar, Mar 15 1996

nonn

Number of maximal independent sets in rooted plane trees on n nodes. - Olivier Gérard, Jul 05 2001

G. C. Greubel, Table of n, a(n) for n = 1..1000
Martin Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
A. Mironov and A. Morozov, Algebra of quantum C-polynomials, arXiv:2009.11641 [hep-th], 2020.
Index entries for sequences related to rooted trees

Cf. A000108.

series(1-x/RootOf(Z-_Z^2-_Z^3+_Z^4-x), x=0,20); # _Mark van Hoeij, May 28 2013

Rest[CoefficientList[1-x/InverseSeries[Series[x-x^2-x^3+x^4, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[Sum[Binomial[n + k, k]/(n + k)*Sum[(Binomial[j, n - k - j + 1]*Binomial[k, j]*(-1)^(n + k - j + 1)), {j, 0, k}], {k, 1, n}] + CatalanNumber[n], {n, 0, 50}] (* G. C. Greubel, Feb 15 2017 *)

a(n):=sum(binomial(n+k,k)/(n+k)*sum(binomial(j,n-k-j+1)*binomial(k,j)*(-1)^(n+k-j+1),j,0,k),k,1,n)+1/(n+1)*binomial(2*n,n); /* Vladimir Kruchinin, Nov 13 2014 */

my(x='x+O('x^66)); Vec(1-x/serreverse(x-x^2-x^3+x^4)) \\ Joerg Arndt, May 28 2013

a(n+1) = Sum_{k = 1..n} ( binomial(n+k,k)/(n+k)*Sum_{j = 0..k} ( binomial(j,n-k-j+1)*binomial(k,j)*(-1)^(n+k-j+1) ) ) + C(n), where C(n) is a Catalan number. - Vladimir Kruchinin, Nov 13 2014
Recurrence: 16*(n-1)*n*(2*n-3)*(17*n^2 - 81*n + 96)*a(n) = (n-1)*(1819*n^4 - 14124*n^3 + 40377*n^2 - 50320*n + 23040)*a(n-1) + 8*(2*n-5)*(4*n-11)*(4*n-9)*(17*n^2 - 47*n + 32)*a(n-2). - Vaclav Kotesovec, Nov 14 2014
Asymptotics (Klazar, 1997): a(n) ~ sqrt(5731-4635/sqrt(17)) * ((107+51*sqrt(17))/64)^n / (256 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 14 2014

A348410 Number of nonnegative integer solutions to n = Sum_{i=1..n} (a_i + b_i), with b_i even.

Original entry on oeis.org

1, 1, 5, 19, 85, 376, 1715, 7890, 36693, 171820, 809380, 3830619, 18201235, 86770516, 414836210, 1988138644, 9548771157, 45948159420, 221470766204, 1069091485500, 5167705849460, 25009724705460, 121171296320475, 587662804774890, 2852708925078675, 13859743127937876

Offset: 0

César Eliud Lozada, Oct 17 2021

Suppose n objects are to be distributed into 2n baskets, half of these white and half black. White baskets may contain 0 or any number of objects, while black baskets may contain 0 or an even number of objects. a(n) is the number of distinct possible distributions.

			Some examples (semicolon separates white basket from black baskets):
For n=1: {{1 ; 0}} - Total possible ways: 1.
For n=2: {{0, 0 ; 0, 2}, {0, 0 ; 2, 0}, {0, 2 ; 0, 0}, {1, 1 ; 0, 0}, {2, 0 ; 0, 0}} - Total possible ways: 5.

Alois P. Heinz, Table of n, a(n) for n = 0..1441

Cf. A033715, A033716, A034297, A063020, A088218, A234839, A294860, A348474, A351856, A351857, A352373.

b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
      add(b(n-j, t-1)*(1+iquo(j, 2)), j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 17 2021

(* giveList=True produces the list of solutions *)
(* giveList=False gives the number of solutions *)
counter[objects_, giveList_: False] :=
  Module[{n = objects, nb, eq1, eqa, eqb, eqs, var, sol, var2, list},
   nb = n;
   eq1 = {Total[Map[a[#] + 2*b[#] &, Range[nb]]] - n == 0};
   eqa = {And @@ Map[0 <= a[#] <= n &, Range[nb]]};
   eqb = {And @@ Map[0 <= b[#] <= n &, Range[nb]]};
   eqs = {And @@ Join[eq1, eqa, eqb]};
   var = Flatten[Map[{a[#], b[#]} &, Range[nb]]];
   var = Join[Map[a[#] &, Range[nb]], Map[b[#] &, Range[nb]]];
   sol = Solve[eqs, var, Integers];
   var2 = Join[Map[a[#] &, Range[nb]], Map[2*b[#] &, Range[nb]]];
   list = Sort[Map[var2 /. # &, sol]];
   list = Map[StringReplace[ToString[#], {"," -> " ;"}, n] &, list];
   list = Map[StringReplace[#, {";" -> ","}, n - 1] &, list];
   Return[
    If[giveList, Print["Total: ", Length[list]]; list, Length[sol]]];
   ];
(* second program: *)
b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n], Sum[b[n - j, t - 1]*(1 + Quotient[j, 2]), {j, 0, n}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 16 2023, after Alois P. Heinz *)

Conjecture: D-finite with recurrence +7168*n*(2*n-1)*(n-1)*a(n) -64*(n-1)*(1759*n^2-5294*n+5112)*a(n-1) +12*(7561*n^3-75690*n^2+165271*n-101070)*a(n-2) +5*(110593*n^3-743946*n^2+1659971*n-1232778)*a(n-3) +2680*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Oct 19 2021
From Vaclav Kotesovec, Nov 01 2021: (Start)
Recurrence (of order 2): 16*(n-1)*n*(2*n - 1)*(51*n^2 - 162*n + 127)*a(n) = (n-1)*(5457*n^4 - 22791*n^3 + 32144*n^2 - 17536*n + 3072)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(51*n^2 - 60*n + 16)*a(n-2).
a(n) ~ sqrt(3 + 5/sqrt(17)) * (107 + 51*sqrt(17))^n / (sqrt(Pi*n) * 2^(6*n+2)). (End)
From Peter Bala, Feb 21 2022: (Start)
a(n) = [x^n] ( (1 - x)*(1 - x^2) )^(-n). Cf. A234839.
a(n) = Sum_{k = 0..floor(n/2)} binomial(2*n-2*k-1,n-2*k)*binomial(n+k-1,k).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 119*x^5 + ... is the g.f. of A063020.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k.
The o.g.f. A(x) is the diagonal of the bivariate rational function 1/(1 - t/((1-x)*(1-x^2))) and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.
Let F(x) = (1/x)*Series_Reversion( x*(1-x)*(1-x^2) ). Then A(x) = 1 + x*d/dx (log(F(x))). (End)
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(2*n+k-1, k)*binomial(2*n-k-1, n-k). Cf. A352373. - Peter Bala, Jun 05 2024

More terms from Alois P. Heinz, Oct 17 2021

A365752 Expansion of (1/x) * Series_Reversion( x(1+x)(1-x)^4 ).

Original entry on oeis.org

1, 3, 16, 103, 735, 5592, 44452, 364815, 3067558, 26290517, 228819168, 2016953848, 17968790029, 161536295244, 1463535347928, 13349907110367, 122499957767130, 1130001670577730, 10472708110616136, 97468774074103041, 910582642690819351

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Index entries for reversions of series

Cf. A063020, A365751, A365753.
Cf. A365856, A368079.
Cf. A365754, A370105.

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

def A365752(n):
    h = binomial(5*n + 3, n) * hypergeometric([-n, n + 1], [-5 * n - 3], -1) / (n + 1)
    return simplify(h)
print([A365752(n) for n in range(21)])  # Peter Luschny, Sep 20 2023

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+k,k) * binomial(5*n-k+3,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(4*n-2*k+2,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x) * (1-x)^4 )^(n+1). - Seiichi Manyama, Feb 16 2024

A063030 Reversion of y - y^2 - y^4 + y^5.

Original entry on oeis.org

0, 1, 1, 2, 6, 19, 63, 220, 795, 2942, 11099, 42536, 165126, 647955, 2565946, 10241616, 41158598, 166402323, 676338003, 2761988994, 11327162406, 46631572295, 192638451780, 798316442580, 3317866307145, 13825837134096

Offset: 0

Olivier Gérard, Jul 05 2001

R. J. Mathar, Table of n, a(n) for n = 0..105
Index entries for reversions of series

Cf. A006013, A063020.

Cf. A063026.

CoefficientList[InverseSeries[Series[y - y^2 - y^4 + y^5, {y, 0, 30}], x], x]

a(n)=if(n<1,0,polcoeff(serreverse(x-x^2-x^4+x^5+x*O(x^n)),n))

D-finite with recurrence 1458*n*(n-1)*(n-2)*(2*n-1) *(981649511*n -2631216939)*a(n) -486*(n-1)*(n-2) *(24210415932*n^3 -114067288649*n^2 +155533650884*n -64732315335)*a(n-1) +54*(n-2) *(39787015892*n^4 -313539301751*n^3 +992577496688*n^2 -1613867842189*n +1173502139880)*a(n-2) +(-27607572942679*n^5 +295135536608825*n^4 -1205223186688595*n^3 +2314131935158975*n^2 -2033367943220766*n +619177732684560)*a(n-3) -5*(5*n-21) *(5408009*n +1144402484)*(5*n-19) *(5*n-18)*(5*n-17) *a(n-4)=0. - R. J. Mathar, Mar 21 2022

a(n+1) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,n) * binomial(2*n-3*k,n). - Seiichi Manyama, Sep 26 2023

A365753 Expansion of (1/x) * Series_Reversion( x(1+x)(1-x)^5 ).

Original entry on oeis.org

1, 4, 27, 220, 1984, 19064, 191325, 1981932, 21031965, 227463808, 2498039219, 27782561352, 312281382836, 3541879743840, 40484779373060, 465888833819532, 5393215780225983, 62761359573224612, 733784067570047400, 8615217370731224160, 101533102164551821896

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Index entries for reversions of series

Cf. A063020, A365751, A365752.

Cf. A365755.

a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(6*n-k+4, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+k,k) * binomial(6*n-k+4,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(5*n-2*k+3,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x) * (1-x)^5 )^(n+1). - Seiichi Manyama, Feb 16 2024

A365751 Expansion of (1/x) * Series_Reversion( x(1+x)(1-x)^3 ).

Original entry on oeis.org

1, 2, 8, 38, 201, 1134, 6688, 40734, 254237, 1617572, 10452416, 68408626, 452530659, 3020870352, 20324167488, 137672551630, 938154745773, 6426806842566, 44234352581896, 305743015718028, 2121318029754770, 14769052147618740, 103148538125870880

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Index entries for reversions of series

Cf. A063020, A365752, A365753.

Cf. A352373.

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

def A365751(n):
    h = binomial(4*n + 2, n) * hypergeometric([-n, n + 1], [-4 * n - 2], -1) / (n + 1)
    return simplify(h)
print([A365751(n) for n in range(23)])  # Peter Luschny, Sep 20 2023

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+k,k) * binomial(4*n-k+2,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(3*n-2*k+1,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x) * (1-x)^3 )^(n+1). - Seiichi Manyama, Feb 16 2024

A365854 Expansion of (1/x) * Series_Reversion( x(1+x)^2(1-x)^3 ).

Original entry on oeis.org

1, 1, 4, 13, 55, 232, 1052, 4869, 23206, 112519, 554560, 2767336, 13959941, 71060356, 364569352, 1883143669, 9785481498, 51118097686, 268294595396, 1414106565611, 7481787454031, 39721596068000, 211549545257760, 1129912319370600, 6050931114958080

Offset: 0

Seiichi Manyama, Sep 20 2023

nonn

Index entries for reversions of series

Cf. A365855, A365856.
Cf. A063020, A365878, A365880.
Cf. A365751, A370103.

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

def A365854(n):
    h = binomial(2*(2*n + 1), n) * hypergeometric([-n, 2*(n + 1)], [-2*(2*n + 1)], -1) / (n + 1)
    return simplify(h)
print([A365854(n) for n in range(25)])  # Peter Luschny, Sep 20 2023

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(2*n+k+1,k) * binomial(4*n-k+2,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(2*n-2*k,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^2 * (1-x)^3 )^(n+1). - Seiichi Manyama, Feb 16 2024

A133656 Number of below-diagonal paths from (0,0) to (n,n) using steps (1,0), (0,1) and (2k-1,1), k a positive integer.

Original entry on oeis.org

1, 2, 6, 23, 99, 456, 2199, 10962, 56033, 292094, 1546885, 8299058, 45010492, 246377362, 1359339710, 7551689783, 42206697209, 237156951618, 1338917298708, 7591380528489, 43207023511013, 246773061257046, 1413889039642479, 8124356140582768, 46807462792903984

Offset: 0

Brian Drake, Sep 20 2007

			a(4) = 99 since there are 90 Schroeder paths (A006318) from (0,0) to (4,4) plus DNNEN, DNENN, DENNN, DdNN, DNdN, DNNd, EDNNN, ENDNN and dDNN, where E=(1,0), N=(0,1), D=(3,1) and d=(1,1).

Alois P. Heinz, Table of n, a(n) for n = 0..1276 (first 51 terms from Brian Drake)
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

Cf. A006318, A064641, A052709, A063020, A348474.

Row sums of A201080.

A:=series(RootOf(1+_Z*(x-1)+_Z^2*(x-x^2)+_Z^3*x^2-_Z^4*x^3), x, 21): seq(coeff(A,x,i), i=0..20);

a[n_] := Sum[Binomial[n+k, n] * Sum[Binomial[j, -n - 3k + 2j - 2]* (-1)^(n+k-j+1) * Binomial[n+k+1, j], {j, 0, k+n+1}], {k, 0, n}]/(n+1);
a /@ Range[0, 24] (* Jean-François Alcover, Oct 06 2019, after Vladimir Kruchinin *)

a(n):=sum(binomial(n+k,n)*sum(binomial(j,-n-3*k+2*j-2)*(-1)^(n+k-j+1) *binomial(n+k+1,j),j,0,k+n+1),k,0,n)/(n+1); /* Vladimir Kruchinin, Oct 11 2011 */

G.f. g(x) satisfies: g(x) = 1 + x*g(x)^2+x*g(x)/(1-x^2*g(x)^2).
a(n) = sum(k=0..n, binomial(n+k,n)*sum(j=0..k+n+1, binomial(j,-n-3*k+2*j-2) *(-1)^(n+k-j+1)*binomial(n+k+1,j)))/(n+1). - Vladimir Kruchinin, Oct 11 2011
From Peter Bala, Feb 22 2022: (Start)
G.f. g(x) = (1/x)*series reversion of x*(1 + x)*(1 - x)^2/(1 + x - x^2).
It appears that 1 + x*g'(x)/g(x) = 1 + 2*x + 8*x^2 + 41*x^3 + 220*x^4 + ... is the g.f. of A348474. (End)

A201079 Irregular triangle read by rows: number of {0,2,4,6...}-shifted Schroeder paths of length n and area k.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 4, 6, 7, 7, 5, 0, 0, 1, 2, 4, 7, 11, 14, 18, 20, 19, 15, 8, 0, 0, 1, 2, 4, 8, 12, 19, 26, 35, 43, 52, 57, 61, 57, 46, 30, 13, 0, 0, 0, 1, 2, 4, 8, 13, 21, 32, 45, 61, 81, 101, 125, 146, 167, 183, 194, 191, 178, 146, 103, 58, 21, 0, 0, 0

Offset: 0

N. J. A. Sloane, Nov 26 2011

			Triangle begins
1
1
1 2 0
1 2 3 3 0
1 2 4 6 7 7 5 0 0
1 2 4 7 11 14 18 20 19 15 8 0 0
1 2 4 8 12 19 26 35 43 52 57 61 57 46 30 13 0 0 0
...

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

Row sums give A063020. Rows converge to A015128.

Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201076 {0,2}, A201080 {0,1,3,5...}, A201159 {0,1,2}.

max = 8; s0 = Range[2, max, 2];
gf = Expand /@ FixedPoint[With[{g = Normal@#}, 1 + q x g (g /. {x :> q^2 x}) + Sum[q^(j^2 - j) x^j Product[g /. {x :> q^(2 i - 2) x}, {i, j}], {j, s0}] + O[x]^max] &, 0];
Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 02 2024 *)

Name and rows 3 and 5 corrected and row 7 added by Andrey Zabolotskiy, Jan 02 2024

A366041 Expansion of (1/x) * Series_Reversion( x(1-x)(1-x^4) ).

Original entry on oeis.org

1, 1, 2, 5, 15, 48, 160, 549, 1930, 6919, 25200, 92976, 346757, 1305140, 4951216, 18912245, 72675114, 280761670, 1089800270, 4248149795, 16623209911, 65273370720, 257115465600, 1015719256200, 4023178881540, 15974388769653, 63570826294760

Offset: 0

Seiichi Manyama, Sep 26 2023

nonn

Cf. A006013, A063020, A063030, A366042.

Cf. A366023.

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

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

cp's OEIS Frontend

A007858 G.f. is 1 - 1/f(x), where f(x) = 1+x+3*x^2+9*x^3+32*x^4+... is 1/x times g.f. for A063020.

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A348410 Number of nonnegative integer solutions to n = Sum_{i=1..n} (a_i + b_i), with b_i even.

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A365752 Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^4 ).

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A063030 Reversion of y - y^2 - y^4 + y^5.

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A365753 Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^5 ).

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A365751 Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^3 ).

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A365854 Expansion of (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^3 ).

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A133656 Number of below-diagonal paths from (0,0) to (n,n) using steps (1,0), (0,1) and (2k-1,1), k a positive integer.

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A007858 G.f. is 1 - 1/f(x), where f(x) = 1+x+3x^2+9x^3+32*x^4+... is 1/x times g.f. for A063020.

A365752 Expansion of (1/x) * Series_Reversion( x(1+x)(1-x)^4 ).

A365753 Expansion of (1/x) * Series_Reversion( x(1+x)(1-x)^5 ).

A365751 Expansion of (1/x) * Series_Reversion( x(1+x)(1-x)^3 ).

A365854 Expansion of (1/x) * Series_Reversion( x(1+x)^2(1-x)^3 ).

A366041 Expansion of (1/x) * Series_Reversion( x(1-x)(1-x^4) ).