A051286 -id:A051286 - cp's OEIS frontend

A377148 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(k,n-k)^2.

1, 4, 14, 60, 225, 796, 2764, 9304, 30580, 98700, 313422, 981548, 3037473, 9301620, 28222000, 84927760, 253699285, 752863840, 2220831160, 6515581600, 19021079866, 55276625304, 159967084164, 461150383400, 1324652146775, 3792447499916, 10824189204014

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A051286, A182884, A377145, A377152, A377153, A377158, A377159.
Cf. A377149, A377150.
Cf. A089627.

[&+[Binomial(k+3,3)*Binomial(k, n-k)^2: k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, May 12 2025

Table[Sum[Binomial[k+3,3]*Binomial[k, n-k]^2,{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 12 2025 *)

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

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=3, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (1-x-x^2) * ((1-x-x^2)^2 + 6*x^3) / ((1-x-x^2)^2 - 4*x^3)^(7/2).

A377145 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 3, 9, 34, 111, 351, 1103, 3384, 10224, 30536, 90222, 264186, 767663, 2215623, 6356907, 18143300, 51540885, 145801395, 410888595, 1153964520, 3230723826, 9019081038, 25112021154, 69750583164, 193303849531, 534602071341, 1475644537323, 4065845732794

Offset: 0

Seiichi Manyama, Oct 17 2024

nonn

Cf. A051286, A182884, A377148, A377152, A377153, A377158, A377159.
Cf. A377146, A377147.
Cf. A089627.

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

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=2, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: ((1-x-x^2)^2 + 2*x^3) / ((1-x-x^2)^2 - 4*x^3)^(5/2).

A078698 Number of ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

Original entry on oeis.org

1, 2, 20, 396, 14976, 907200, 79315200, 9551001600, 1513528934400, 305106949324800, 76296489615360000, 23175289163980800000, 8404709419090575360000, 3587225703492542791680000, 1779970753996760560435200000, 1016036270188884847558656000000, 661106386935312429191528448000000

Offset: 1

Hugo Pfoertner, Dec 18 2002

nonn

The lace is "directed": reversing the order of eyelets along the path counts as a different solution. It must begin and end at the extreme pair of eyelets,

			a(3) = 20: label the eyelets 1,2,3 from front to back on the left side then 4,5,6 from back to front on the right side. The lacings are: 124356 154326 153426 142536 145236 135246 and the following and their mirror images: 125346 124536 125436 152346 153246 152436 154236.
Examples for n=2,3,4 can be found following the FORTRAN program at given link.

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 494.

A. Khrabrov and K. Kokhas, Points on a line, shoelace and dominoes, arXiv:1505.06309 [math.CO], (23-May-2015).
Hugo Pfoertner, Fortran program.
Hugo Pfoertner, Results for n=1..4.
Hugo Pfoertner, Results for n=5.
Index entries for sequences related to shoe lacings

Cf. A078700, A078702, A078602.

Fortran
```
c Program provided at Pfoertner link
```

a[n_] := (n-1)!^2 Sum[Binomial[n-k, k]^2, {k, 0, n/2}];
Array[a, 17] (* Jean-François Alcover, Jul 20 2018 *)

Conjecture: a(n) = (n-1)!^2*A051286(n). - Vladeta Jovovic, Sep 14 2005 (correct, see the Khrabrov/Kokhas reference, Joerg Arndt, May 26 2015)

Terms a(9) and beyond (using A051286) from Joerg Arndt, May 26 2015

A181546 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)^4.

Original entry on oeis.org

1, 1, 2, 17, 83, 338, 1923, 11553, 63028, 359203, 2172469, 13026034, 78106885, 478415635, 2957675956, 18321372721, 114301292581, 718253640196, 4531427831111, 28699590926291, 182566373639352, 1165539703613397

Offset: 0

Paul D. Hanna, Oct 29 2010

nonn

Conjecture: Given F(n,L) = Sum_{k=0..[n/2]} C(n-k,k)^L, then lim_{n->oo} F(n+1,L)/F(n,L) = (Fibonacci(L)*sqrt(5) + Lucas(L))/2 for L>=0 where Fibonacci(n) = A000045(n) and Lucas(n) = A000032(n).
For this sequence (L=4): lim_{n->oo} a(n+1)/a(n) = (3*sqrt(5)+7)/2 = 6.8541...
Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - w) - (x*y*z*w)^2). - Ilya Gutkovskiy, Apr 23 2025

			G.f. A(x) = 1 + x + 2*x^2 + 17*x^3 + 83*x^4 + 338*x^5 + 1923*x^6 +...
The terms begin:
a(0) = a(1) = 1^4;
a(2) = 1^4 + 1^4 = 2;
a(3) = 1^4 + 2^4 = 17;
a(4) = 1^4 + 3^4 + 1^4 = 83;
a(5) = 1^4 + 4^4 + 3^4 = 338;
a(6) = 1^4 + 5^4 + 6^4 + 1^4 = 1923;
a(7) = 1^4 + 6^4 + 10^4 + 4^4 = 11553; ...

Seiichi Manyama, Table of n, a(n) for n = 0..1202
C. Banderier, P. Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Disc. Appl. Math. 160 (18) (2012) 2542-2554. Table 1.

Cf. variants: A181545, A181547, A051286.

Cf. A000032, A000045.

Table[Sum[Binomial[n-k,k]^4,{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, May 22 2021 *)

PARI
```
{a(n)=sum(k=0,n\2,binomial(n-k,k)^4)}
```

A185828 Half the number of n X 2 binary arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

Original entry on oeis.org

1, 3, 10, 23, 61, 162, 421, 1103, 2890, 7563, 19801, 51842, 135721, 355323, 930250, 2435423, 6376021, 16692642, 43701901, 114413063, 299537290, 784198803, 2053059121, 5374978562, 14071876561, 36840651123, 96450076810, 252509579303

Offset: 1

R. H. Hardin, Feb 05 2011

nonn

Column 2 of A185835.

			Some solutions for 4 X 2 with a(1,1)=0:
  0 0   0 1   0 0   0 0   0 1   0 0   0 0   0 0   0 0   0 0
  1 1   0 1   0 1   1 1   0 1   1 0   0 1   1 1   1 0   0 1
  0 1   0 0   0 1   0 1   1 0   1 0   1 1   1 1   1 1   0 1
  0 0   1 1   0 0   0 1   1 0   0 0   0 0   0 0   0 0   0 1
The logarithmic g.f. begins:
L(x) = x + 3*x^2/2 + 10*x^3/3 + 23*x^4/4 + 61*x^5/5 + 162*x^6/6 + ..., where
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 26*x^5 + 63*x^6 + ... + A051286(n)*x^n/n + ... - _Paul D. Hanna_, Mar 19 2011

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A051286 (exp), A180662 (Fi1).

a := proc(n): n*add(binomial(2*n-2*k, 2*k)/(n-k), k=0..n-1) end: seq(a(n), n=1..28); # Johannes W. Meijer, Jun 18 2018

{a(n)=n*sum(k=0, n-1, binomial(2*n-2*k, 2*k)/(n-k))} /* Paul D. Hanna, Mar 19 2011 */

{a(n)=n*polcoeff(-log( (1+x+x^2)*(1-3*x+x^2) +x*O(x^n))/2, n)} /* Paul D. Hanna, Mar 19 2011 */

Empirical: a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4).
a(n) = n*Sum_{k=0..n-1} C(2n-2k, 2k)/(n-k). - Paul D. Hanna, Mar 19 2011
L.g.f.: Sum_{n>=1} a(n)*x^n/n = -log((1+x+x^2)*(1-3*x+x^2))/2. - Paul D. Hanna, Mar 19 2011
Logarithmic derivative of A051286, which is the Whitney number of level n of the lattice of the ideals of the fence of order 2n. - Paul D. Hanna, Mar 19 2011
Empirical g.f.: x*(1+x+3*x^2-2*x^3)/(1+x+x^2)/(1-3*x+x^2). - Colin Barker, Feb 22 2012
Empirical: a(n) = Sum_{k=0..floor(n/2)} A084534(n, 2*k). - Johannes W. Meijer, Jun 17 2018
Empirical: a(n) = A100886(2n). - Wojciech Florek, Jan 26 2020

A377152 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 5, 20, 95, 400, 1561, 5915, 21610, 76585, 265075, 898622, 2992235, 9810290, 31727815, 101379175, 320464280, 1003259080, 3113576320, 9586763720, 29305985800, 88997753446, 268642069750, 806394498200, 2408144329250, 7157177344225, 21177323087891

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Robert Israel, Table of n, a(n) for n = 0..2357

Cf. A051286, A182884, A377145, A377148, A377153, A377158, A377159.

Cf. A001873, A089627.

f:= proc(n) local k; add(binomial(k+4,4)*binomial(k,n-k)^2,k=0..n) end proc:
map(f, [$0..50]); # Robert Israel, Dec 05 2024

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

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=4, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (Sum_{k=0..2} A089627(4,k) * (1-x-x^2)^(4-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(9/2).

A384747 Triangle read by rows: T(n,k) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,..,k}, and no nodes have the same weight as their parent node.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 5, 6, 0, 1, 11, 15, 16, 0, 1, 26, 39, 43, 44, 0, 1, 63, 110, 123, 127, 128, 0, 1, 153, 308, 358, 371, 375, 376, 0, 1, 376, 869, 1046, 1096, 1109, 1113, 1114, 0, 1, 931, 2499, 3098, 3278, 3328, 3341, 3345, 3346, 0, 1, 2317, 7238, 9283, 9904, 10084, 10134, 10147, 10151, 10152

Offset: 0

John Tyler Rascoe, Jun 09 2025

			Triangle begins:
    k=0  1    2     3     4     5     6     7     8     9
 n=0 [1]
 n=1 [0, 1]
 n=2 [0, 1,   2]
 n=3 [0, 1,   5,    6]
 n=4 [0, 1,  11,   15,   16]
 n=5 [0, 1,  26,   39,   43,   44]
 n=6 [0, 1,  63,  110,  123,  127,  128]
 n=7 [0, 1, 153,  308,  358,  371,  375,  376]
 n=8 [0, 1, 376,  869, 1046, 1096, 1109, 1113, 1114]
 n=9 [0, 1, 931, 2499, 3098, 3278, 3328, 3341, 3345, 3346]
...
T(3,3) = 6 counts:
  o    o    o      o        o        __o__
  |    |    |     / \      / \      /  |  \
 (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
       |    |
      (1)  (2)

Cf. A051286 (column k=2), A382096 (column k=3), A384748 (main diagonal).

Cf. A000108, A002212, A143330, A384613, A384685.

b(i,j,k,N) = {if(k>N,1, 1/( 1  - sum(u=1,j, if(u==i,0,x^u * b(u,j,k+1,N-u+1)))))}
Gx(k,N) = {my(x='x+O('x^(N+1))); Vec(1/(1 - sum(i=1,k, b(i,k,1,N)*x^i)))}
T(max_row) = { my( N = max_row+1, v = vector(N, i, if(i==1, 1, 0))~); for(k=1, N, v=matconcat([v, Gx(k,N)~])); vector(N, n, vector(n, k, v[n, k]))}
T(9)

T(n,k) = T(n,n) for k > n.

A077419 Largest Whitney number of Fibonacci lattices J(Z_n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 7, 11, 17, 26, 40, 63, 97, 153, 238, 376, 587, 931, 1458, 2317, 3640, 5794, 9124, 14545, 22951, 36631, 57904, 92512, 146461, 234205, 371281, 594169, 943045, 1510192, 2399460, 3844787, 6114555, 9802895, 15603339, 25027296

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

A051286 and A051291, interleaved. a(n) is the maximal element in the n-th row of A079487 or A123245 and in the (n+2)-th row of A078807 or A078808. - Andrey Zabolotskiy, Sep 21 2017

Emanuele Munarini, Mar 05 2007, Table of n, a(n) for n = 0..100
Brian Kent, Sarah Racz, and Sanjit Shashi, Scrambling in quantum cellular automata, arXiv:2301.07722 [quant-ph], 2023.
E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

with(FormalPowerSeries): with(LREtools): # requires Maple 2022
gf:= (1 + 2*x + 2*x^4 - x^6 - (1-x^2)*sqrt(1 - 2*x^2 - x^4 - 2*x^6 + x^8))/(2*x*sqrt(1 - 2*x^2 - x^4 - 2*x^6 + x^8));
re:= FindRE(gf,x,a(n));
inits:= {seq(a(i-1)=[1,1,1,2,2,3,5,7,11,17,26,40,63,97, 153][i],i=1..14)};
rm:=  (n+1)*a(n) +(n-2)*a(n-1) +2*(-n+1)*a(n-2) +2*(-n+1)*a(n-3) +(-n-3)*a(n-4) +(-n+8)*a(n-5) +2*(-n+6)*a(n-6) +2*(-n+7)*a(n-7) +(n-9)*a(n-8) +(n-10)*a(n-9)=0;
minre:= MinimalRecurrence(re, a(n), inits); minrm:= MinimalRecurrence(rm, a(n), inits); # shows that Mathar's recurrence is equivalent
f:= REtoproc(re,a(n),inits); seq(f(n),n=0..40); # Georg Fischer, Oct 22 2022

gf[x_] = (1 + 2 x + 2 x^4 - x^6 - (1 - x^2) Sqrt[1 - 2 x^2 - x^4 - 2 x^6 + x^8])/(2 x Sqrt[1 - 2 x^2 - x^4 - 2 x^6 + x^8]);
Table[SeriesCoefficient[gf[x], {x, 0, n}], {n, 0, 40}] (* Hugo Pfoertner, Oct 22 2022 *)

G.f.: (1 + 2 x + 2 x^4 - x^6 - (1-x^2) sqrt(1 - 2 x^2 - x^4 - 2 x^6 + x^8) )/(2x sqrt(1 - 2 x^2 - x^4 - 2 x^6 + x^8)). - Emanuele Munarini, Mar 05 2007
a(n) ~ phi^(n+2) / (5^(1/4) * sqrt(2*Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 22 2017
D-finite with recurrence: (n+1)*a(n) +(n-2)*a(n-1) +2*(-n+1)*a(n-2) +2*(-n+1)*a(n-3) +(-n-3)*a(n-4) +(-n+8)*a(n-5) +2*(-n+6)*a(n-6) +2*(-n+7)*a(n-7) +(n-9)*a(n-8) +(n-10)*a(n-9)=0. - R. J. Mathar, Nov 19 2019

More terms from Emanuele Munarini, Mar 05 2007

A108488 Expansion of 1/sqrt(1 -2x -3x^2 -4x^3 +4x^4).

Original entry on oeis.org

1, 1, 3, 9, 23, 69, 203, 601, 1815, 5493, 16731, 51225, 157367, 485093, 1499499, 4646233, 14427095, 44880981, 139849979, 436419737, 1363713015, 4266417221, 13362194571, 41891406681, 131452430999, 412835452213, 1297543367835

Offset: 0

Paul Barry, Jun 04 2005

In general, Sum_{k=0..n} C(n-k,k)^2*a^k*b^(n-k) has the expansion 1/sqrt(1 -2*b*x -(2*a*b -b^2)*x^2 -2*a*b^2*x^3 +(a*b)^2*x^4).

Diagonal of the rational function 1 / ((1 - x)*(1 - y) - 2*(x*y)^2). - Ilya Gutkovskiy, Apr 23 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A051286, A108480.

Table[Sum[Binomial[n-k,k]^2*2^k,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 24 2013 *)
CoefficientList[Series[1/Sqrt[1-2x-3x^2-4x^3+4x^4],{x,0,30}],x] (* Harvey P. Dale, Apr 06 2023 *)

{a(n)=polcoeff( exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k) * 2^k * x^k) *x^m/m) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 31 2014

a(n) = Sum_{k=0..n} C(n-k, k)^2*2^k.
a(n) ~ ((4*sqrt(2)-1)/62)^(1/4) * (1+2*sqrt(2)+sqrt(1+4*sqrt(2)))^(n+1) /(sqrt(Pi*n)*2^(n+2)). - Vaclav Kotesovec, Jul 24 2013
D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +3*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3) +4*(n-2)*a(n-4)=0. - R. J. Mathar, Aug 06 2013
G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * 2^k * x^k ). - Paul D. Hanna, Aug 31 2014

A174882 A (3/2,-1) Somos-4 sequence.

Original entry on oeis.org

1, 1, -2, -8, -16, -16, 32, 128, 256, 256, -512, -2048, -4096, -4096, 8192, 32768, 65536, 65536, -131072, -524288, -1048576, -1048576, 2097152, 8388608, 16777216, 16777216, -33554432, -134217728, -268435456, -268435456

Offset: 0

Paul Barry, Mar 31 2010

Hankel transform of A051286. a(n+2) = -(-1)^floor(n/4) * 2^A098181(n).

			G.f. = 1 + x - 2*x^2 - 8*x^3 - 16*x^4 - 16*x^5 + 32*x^6 + 128*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,-16).

Cf. A051286, A098181.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-2*x)*(4*x^2+3*x+1)/(1+16*x^4))) // G. C. Greubel, Feb 21 2018

a[ n_] := (-1)^Quotient[n + 2, 4] 2^(n - Mod[ Quotient[n + 1, 2], 2]); (* Michael Somos, Sep 18 2014 *)
CoefficientList[Series[(1-2*x)*(4*x^2+3*x+1)/(1+16*x^4), {x,0,50}], x] (* G. C. Greubel, Feb 21 2018 *)

{a(n) = (-1)^((n+2) \ 4) * 2^(n - ((n+1) \ 2 % 2))}; /* Michael Somos, Jan 06 2011 */

x='x+O('x^30); Vec((1-2*x)*(4*x^2+3*x+1)/(1+16*x^4)) \\ G. C. Greubel, Feb 21 2018

a(n) = ((3/2)*a(n-1)*a(n-3) - a(n-2)^2)/a(n-4), n>3.
a(-n) = a(n-1) / 2^(2*n - 1) for all n in Z. - Michael Somos, Jan 06 2011
0 = a(n)*(+2*a(n+4)) + a(n+1)*(-3*a(n+3)) + a(n+2)*(+2*a(n+2)) for all n in Z. - Michael Somos, Sep 18 2014
a(n+4) = -16 * a(n) for all n in Z. - Michael Somos, Sep 02 2015
G.f.: -(2*x-1)*(4*x^2+3*x+1)/(1+16*x^4) . - R. J. Mathar, Aug 18 2017

cp's OEIS Frontend

A377148 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(k,n-k)^2.

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A377145 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(k,n-k)^2.

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A078698 Number of ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

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A181546 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)^4.

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A185828 Half the number of n X 2 binary arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

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A377152 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(k,n-k)^2.

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A384747 Triangle read by rows: T(n,k) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,..,k}, and no nodes have the same weight as their parent node.

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A108488 Expansion of 1/sqrt(1 -2x -3x^2 -4x^3 +4x^4).