A000346 -id:A000346 - cp's OEIS frontend

A069731 Number of unicursal planar maps with n edges rooted at a vertex of odd valency (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

1, 5, 28, 168, 1056, 6864, 45760, 311168, 2149888, 15049216, 106502144, 760729600, 5477253120, 39710085120, 289650032640, 2124100239360, 15651264921600, 115819360419840, 860372391690240

Offset: 1

Valery A. Liskovets, Apr 07 2002

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069724, A069720, A000108, A003645.

Cf. A000346, A001006, A136576.

Z:=-(1-4*z-sqrt(1-4*z))/sqrt(1-4*z)/64: Zser:=series(Z, z=0, 32): seq(coeff(Zser*2^(n+1), z, n), n=3..24); # Zerinvary Lajos, Jan 01 2007

Table[2^(n-2) CatalanNumber[n+1], {n, 1, 19}] (* Jean-François Alcover, Aug 28 2019 *)

a(n) = 2^(n-2)*C_(n+1), where C_n stands for the Catalan numbers (A000108).
a(n) = A003645(n+2)/4.
D-finite with recurrence: 4*(2*n+1)*a(n-1) - (n+2)*a(n) = 0, a(1) = 1. - Georg Fischer, May 23 2021
From Peter Bala, Apr 29 2024: (Start)
a(n) = Sum_{k = 0..n} binomial(n, 2*k)*Catalan(k)*4^(n-k-1).
O.g.f.: A(x) = (1 - 4*x - 8*x^2 - sqrt(1 - 8*x))/(32*x^2).
A(x) = series reversion of x*c(-x)/(1 + 4*x), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and c(-x)/(1 + 4*x) is the g.f. of (-1)^n*A000346(n). (End)

A187926 Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/2).

Original entry on oeis.org

1, 4, 1, 16, 5, 1, 64, 22, 6, 1, 256, 93, 29, 7, 1, 1024, 386, 130, 37, 8, 1, 4096, 1586, 562, 176, 46, 9, 1, 16384, 6476, 2380, 794, 232, 56, 10, 1, 65536, 26333, 9949, 3473, 1093, 299, 67, 11, 1, 262144, 106762, 41226, 14893, 4944, 1471, 378, 79, 12, 1, 1048576, 431910, 169766, 63004, 21778, 6885, 1941, 470, 92, 13, 1

Offset: 0

Emanuele Munarini, Mar 16 2011

Row sums are A000346.

			Triangle begins:
1,
4,1,
16,5,1,
64,22,6,1,
256,93,29,7,1,
1024,386,130,37,8,1,
4096,1586,562,176,46,9,1,
16384,6476,2380,794,232,56,10,1,
65536,26333,9949,3473,1093,299,67,11,1

Vincenzo Librandi, Table of n, a(n) for n = 0..5150

Cf. A000346

Table[Sum[Binomial[n + i, n]2^(n - k - i), {i, 0, n - k}], {n, 0, 8}, {k, 0, 8}]//MatrixForm

create_list(sum(binomial(n+i,n)*2^(n-k-i),i,0,n-k),n,0,10,k,0,n);

a(n,k) = sum(binomial(n+i,n)*2^(n-k-i),i=0..n-k)

Recurrence: a(n+1,k+1) = a(n,k) + a(n,k+1) + a(n,k+2) + ... + a(n,n).

A296771 Row sums of A050157.

Original entry on oeis.org

1, 3, 13, 58, 257, 1126, 4882, 20980, 89497, 379438, 1600406, 6720748, 28117498, 117254268, 487589572, 2022568168, 8371423177, 34581780478, 142605399982, 587138954428, 2413944555742, 9911778919348, 40650232625212, 166534680737368, 681576405563722

Offset: 0

Peter Luschny, Dec 21 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..1657

Cf. A000346, A037965, A050157, A296770.

A296771 := n -> add(binomial(2*n, n) - binomial(2*n, n+k+1), k=0..n):
seq(A296771(n), n=0..24);

a[n_] := 4^n ((n - 1/2)! (2 n + 3)/(2 Sqrt[Pi] n!) - 1/2);
Table[a[n], {n, 0, 24}]

a(n) = sum(k=0, n, binomial(2*n, n) - binomial(2*n, n+k+1)) \\ Iain Fox, Dec 21 2017

a(n) = Sum_{k=0..n} (binomial(2*n, n) - binomial(2*n, n+k+1)).
a(n) = 2^(2*n-1)*(((n-1/2)!*(2*n+3))/(sqrt(Pi)*n!) - 1).
a(n) ~ 4^n*(sqrt(n/Pi) - 1/2).
a(n) = A037965(n+1) - A000346(n-1) for n >= 1.
From Robert Israel, Dec 21 2017: (Start)
a(n) = (n+3/2)*binomial(2*n,n) - 2^(2*n-1).
G.f.: (3/2-4*x)*(1-4*x)^(-3/2) - (1/2)*(1-4*x)^(-1).
64*(n+1)*(2*n+1)*a(n)-8*(2*n+3)*(5*n+4)*a(n+1)+2*(n+2)*(8*n+11)*a(n+2)-(n+3)*(n+2)*a(n+3)=0. (End)

A303602 a(n) = Sum_{k = 0..n} kbinomial(2n+1, k).

Original entry on oeis.org

0, 3, 25, 154, 837, 4246, 20618, 97140, 447661, 2028478, 9070110, 40122028, 175913250, 765561564, 3310623412, 14238676712, 60949133949, 259809601870, 1103420316566, 4670886541308, 19714134528598, 82985455688276, 348481959315660, 1460179866076504, 6106070639175122

Offset: 0

Bruno Berselli, May 09 2018

Second bisection of A185251; the first bisection is A002699.

The terms are not congruent to 5 (mod 6).

Cf. A000346, A002699, A005408, A047226, A164991, A185251.

seq(add(k*binomial(2*n+1,k),k=0..n),n=0..24); # Paolo P. Lava, May 10 2018

Table[Sum[k Binomial[2 n + 1, k], {k, 0, n}], {n, 0, 30}]
CoefficientList[Series[(1 + 4*x - Sqrt[1 - 4*x]) / (2*(1 - 4*x)^2), {x, 0, 25}], x] (* Vaclav Kotesovec, May 10 2018 *)

a(n)=(2*n+1)*(4^n-binomial(2*n,n))/2 \\ Charles R Greathouse IV, Oct 23 2023

[(2*n+1)*(4^n-binomial(2*n,n))/2 for n in (0..30)]

E.g.f.: ((1 + 8*x)*exp(2*x) - (1 + 4*x)*I_0(2*x) - 4*x*I_1(2*x))*exp(2*x)/2, where I_m(.) is the modified Bessel function of the first kind.
From Vaclav Kotesovec, May 10 2018: (Start)
G.f.: (1 + 4*x - sqrt(1 - 4*x)) / (2*(1 - 4*x)^2).
D-finite with recurrence: n*(2*n-1)*a(n) = 2*(2*n+1)*(4*n-3)*a(n-1) - 8*(2*n-1)*(2*n+1)*a(n-2). (End)
a(n) = (2*n + 1)*(4^n - binomial(2*n, n))/2.
a(n+1) - 4*a(n) = A164991(2*n+3).

A038602 One half of convolution of central binomial coefficients A000984(n) with A000984(n+2), n >= 0.

Original entry on oeis.org

3, 16, 73, 316, 1334, 5552, 22901, 93892, 383290, 1559680, 6331098, 25649976, 103758828, 419195552, 1691825933, 6822051092, 27488564498, 110691186272, 445487285678, 1792047789512, 7205785665908, 28963557761312

Offset: 0

Wolfdieter Lang

Also convolution of A000346 with Catalan numbers but with C(0)=1 replaced by 3

Fung Lam, Table of n, a(n) for n = 0..1500

Cf. A000108, A000346.

CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x)*((1-Sqrt[1-4*x])/(2*x)+2)/(1-4*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 28 2014 *)

a(n) = 2^(2*n+3)-(3*n+5)*C(n+1), C(n): Catalan numbers A000108.
G.f.: c(x)*(c(x)+2)/(1-4*x), where c(x) is G.f. for Catalan numbers.
a(n) ~ 2^(2*n+3) * (1-3/(2*sqrt(Pi*n))).  - Vaclav Kotesovec, Mar 28 2014
Recurrence: n*(n+2)*a(n) = 2*(4*n^2 + 5*n - 1)*a(n-1) - 8*(n+1)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Mar 28 2014

A058893 Triangle arising in enumeration of maps on projective plane.

Original entry on oeis.org

1, 5, 9, 22, 112, 118, 93, 899, 2346, 1773, 386, 5940, 27446, 48426, 28650, 1586, 35138, 247752, 745180, 995290, 484578

Offset: 1

N. J. A. Sloane, Jan 08 2001

			Triangle starts:
1;
5,9;
22,112,118;
93,899,2346,1773;
...

E. A. Bender, E. R. Canfield and R. W. Robinson, The enumeration of maps on the torus and the projective plane, Canad. Math. Bull., 31 (1988), 257-271; see p. 270.

A000346 is first column.

A118447 Number of rooted n-edge one-vertex maps on the Klein bottle (dually: one-face maps).

Original entry on oeis.org

4, 42, 304, 1870, 10488, 55412, 280768, 1379286, 6616360, 31144300, 144367584, 660746892, 2991902704, 13424189160, 59758420736, 264191654758, 1160934273288, 5074150057916, 22071747625120, 95596117130724

Offset: 2

Valery A. Liskovets, May 04 2006

nonn

One-vertex maps on the projective plane are counted by A000346 and one-vertex maps on a non-orientable genus-3 surface by A118448. Such maps are also called bouquets of loops (and their duals are called unicellular maps).

E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
D. M. Jackson and T. I. Visentin, An atlas of the smaller maps in orientable and nonorientable surfaces. CRC Press, Boca Raton, 2001.

((R - 1)^2 (R + 1) (R + 3)/(8 R^5) /. R -> Sqrt[1 - 4x]) + O[x]^22 // CoefficientList[#, x]& // Drop[#, 2]& (* Jean-François Alcover, Aug 28 2019 *)

O.g.f.: (R-1)^2(R+1)(R+3)/8R^5, where R=sqrt(1-4x).
Conjecture: -(n-2)*(n-1)^2*a(n) +2*n*(4*n-5)*(n-2)*a(n-1) -8*n*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jun 22 2016
a(n) ~ n^(3/2) * 2^(2*n-1) / sqrt(Pi) * (1 - sqrt(Pi/n)/2). - Vaclav Kotesovec, Aug 28 2019

A225419 Triangle read by rows: T(n,k) (0 <= k <= n) = binomial(2*n+2,k).

Original entry on oeis.org

1, 1, 4, 1, 6, 15, 1, 8, 28, 56, 1, 10, 45, 120, 210, 1, 12, 66, 220, 495, 792, 1, 14, 91, 364, 1001, 2002, 3003, 1, 16, 120, 560, 1820, 4368, 8008, 11440, 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 1, 20, 190

Offset: 0

Roger L. Bagula, May 07 2013

Row sums are A000346.

			Triangle begins:
1,
1, 4,
1, 6, 15,
1, 8, 28, 56,
1, 10, 45, 120, 210,
1, 12, 66, 220, 495, 792,
1, 14, 91, 364, 1001, 2002, 3003,
...

Roger L. Bagula, a225419nb.txt

Cf. A127673, A062344, A034870, A000346.

Flatten[Table[Table[DifferenceRoot[Function[{y, n}, {(-(2*m + 1) + n) y[n] + n y[1 + n] == 0, y[1] == 1}]][k], {k, 1, m}], {m, 1, 10}]]
Flatten[Table[Binomial[2n+2,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Apr 13 2014 *)

Edited by N. J. A. Sloane, May 11 2013

A271453 Triangle read by rows of coefficients of polynomials C_n(x) = Sum_{k=0..n} (2k)!(x - 1)^(n-k)/((k + 1)!*k!).

Original entry on oeis.org

1, 0, 1, 2, -1, 1, 3, 3, -2, 1, 11, 0, 5, -3, 1, 31, 11, -5, 8, -4, 1, 101, 20, 16, -13, 12, -5, 1, 328, 81, 4, 29, -25, 17, -6, 1, 1102, 247, 77, -25, 54, -42, 23, -7, 1, 3760, 855, 170, 102, -79, 96, -65, 30, -8, 1, 13036, 2905, 685, 68, 181, -175, 161, -95, 38, -9, 1, 45750, 10131, 2220, 617, -113, 356, -336, 256, -133, 47, -10, 1

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

The polynomials C_n(x) have generating function G(x,t) = (1 - sqrt(1 - 4*t))/(2*t*(1 + t - x*t)) = 1 + x*t + (x^2 - x + 2)*t^2 + (x^3 - 2*x^2 + 3*x + 3)*t^3 + ...
C_n(x) can be defined by the recurrence relation C_n(x) = (x - 1)*C_(n-1)(x) + (2n)!/((n + 1)!*n!), C_0(x) = 1 or the equivalent form C_n(x) = (x - 1)*C_(n-1)(x) + C_n(1), C_0(x) = 1.
C_n(x) can be defined as convolution of Catalan numbers and powers of (x - 1).
Discriminants of C_n(x) gives the sequence: 1, 1, -7, -543, 533489, 7080307052, -1318026434480736, -3526797951451513832247, 137992774365121594001729513153, ...
C_n(0) = A032357(n).
C_n(1) = C_n(x) - (x - 1)*C_(n-1)(x) = A000108(n).
C_n(2) = Sum_{m=0..n} C_1(m) = A014137(n).
C_n(3) = A014318(n).
C_n(5) = A000346(n).
C_n(6) = A046714(n).

			Triangle begins:
   1;
   0,  1;
   2, -1,  1;
   3,  3, -2,  1;
  11,  0,  5, -3,  1;
  31, 11, -5,  8, -4,  1;
  ...
The first few polynomials are:
  C_0(x) = 1;
  C_1(x) = x;
  C_2(x) = x^2 -   x   + 2;
  C_3(x) = x^3 - 2*x^2 + 3*x   + 3;
  C_4(x) = x^4 - 3*x^3 + 5*x^2         + 11;
  C_5(x) = x^5 - 4*x^4 + 8*x^3 - 5*x^2 + 11*x + 31;
  ...

G. C. Greubel, Rows n=0..100 of triangle, flattened
Ilya Gutkovskiy, Polynomials C_n(x)
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A130595.

CoefficientList[RecurrenceTable[{c[0] == 1, c[n] == (x - 1) c[n - 1] + CatalanNumber[n]}, c, {n, 11}], x]
T[n_, n_]:= 1; T[n_, 0]:= (-1)^n*Sum[CatalanNumber[k]*(-1)^k, {k, 0, n}]; T[n_, k_]:= T[n - 1, k - 1] - T[n - 1, k]; Table[T[n, k], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 04 2018 *)

{T(n, k) = if(k==n, 1, if(k==0, sum(j=0,n, (-1)^(n-j)*(2*j)!/(j!*(j+1)!)), T(n-1, k-1) - T(n-1, k))) };
for(n=0, 10, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 04 2018

For triangle: T(n,n)=1, T(n,0) = Sum_{k=0..n} (-1)^(n-k)*(2*k)!/(k! * (k+1)!), T(n, k) = T(n-1, k-1) - T(n-1, k). - G. C. Greubel, Nov 04 2018

A306609 a(n) = Sum_{k=0..n} kbinomial(4n+2,2*k).

Original entry on oeis.org

0, 15, 465, 11102, 236997, 4751010, 91474890, 1712391420, 31398038701, 566621243642, 10097483539038, 178113001428004, 3115342162844450, 54103694774702292, 933929099838928692, 16037182307150776056, 274132978890654857853, 4667160114821964359530, 79177297937966956038102, 1338972240005810710258452

Offset: 0

Robert Israel, Feb 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..828
MathOverflow, Specific partial sum of even/odd binomial coefficients

Cf. A000346.

List([0..30],n->Sum([0..n],k->k*Binomial(4*n+2,2*k))); # Muniru A Asiru, Mar 01 2019

f:= n -> (n+1/2)*(16^n-binomial(4*n,2*n)):
map(f, [$0..30]);

Table[Sum[k Binomial[4n+2,2k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jun 14 2024 *)

a(n) = (n+1/2)*(16^n - binomial(4*n,2*n)) = (2*n+1)*A000346(2*n-1).
-512*(4*n + 1)*(86*n + 213)*(3 + 4*n)*(n + 1)*a(n) + 32*(2336*n^4 + 8800*n^3 + 10524*n^2 + 11540*n + 9703)*a(n + 1) - 2*(n + 2)*(544*n^3 - 1072*n^2 + 1138*n + 8055)*a(n + 2) - (2*n + 5)*(26*n - 31)*(n + 3)*(n + 2)*a(n + 3) = 0.
a(n) ~ 16^n * (n - sqrt(n/(2*Pi)) + 1/2).

cp's OEIS Frontend

A069731 Number of unicursal planar maps with n edges rooted at a vertex of odd valency (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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A187926 Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/2).

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A296771 Row sums of A050157.

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A303602 a(n) = Sum_{k = 0..n} k*binomial(2*n+1, k).

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A038602 One half of convolution of central binomial coefficients A000984(n) with A000984(n+2), n >= 0.

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A058893 Triangle arising in enumeration of maps on projective plane.

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A118447 Number of rooted n-edge one-vertex maps on the Klein bottle (dually: one-face maps).

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Formula

A225419 Triangle read by rows: T(n,k) (0 <= k <= n) = binomial(2*n+2,k).

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A303602 a(n) = Sum_{k = 0..n} kbinomial(2n+1, k).

A271453 Triangle read by rows of coefficients of polynomials C_n(x) = Sum_{k=0..n} (2k)!(x - 1)^(n-k)/((k + 1)!*k!).

A306609 a(n) = Sum_{k=0..n} kbinomial(4n+2,2*k).