A051286 -id:A051286 - cp's OEIS frontend

A323769 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n.

1, 1, 2, 9, 83, 1268, 62283, 10296321, 2668655428, 1306416217435, 3055324257386077, 17213278350960504924, 137320554100797006975445, 3087543920644806918694851647, 335732238884967561227813578781572, 61125387696211835948801235842204794881

Offset: 0

Seiichi Manyama, Jan 27 2019

nonn

The limit a(n) / (5^(n/4) * phi^(n*(n+1)) / (2*Pi*n)^(n/2)) does not exist but oscillates between 2 attractors. The value is dependent on the fractional part of n/(sqrt(5)*phi), see graph. - Vaclav Kotesovec, Jan 28 2019

Seiichi Manyama, Table of n, a(n) for n = 0..71
Vaclav Kotesovec, Graph - the asymptotic ratio
Vaclav Kotesovec, Graph - dependence of the limit on the fractional part of n/(sqrt(5)*phi)

Main diagonal of A323767.

Cf. A011973, A051286, A181545, A181546, A181547, A209428, A323768.

Table[Sum[Binomial[n-k,k]^n, {k, 0, n/2}], {n, 0, 15}] (* Vaclav Kotesovec, Jan 27 2019 *)

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

a(n)^(1/n) ~ 5^(1/4) * phi^(n+1) / sqrt(2*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 27 2019

log(a(n)) ~ n*(n*v + w - log(n))/2 with v = 2*log((1 + sqrt(5))/2) and w = log((35 + 15*sqrt(5))/(8*Pi^2))/2, preceding formula recast. - Peter Luschny, Jan 28 2019

A375021 Expansion of 1/sqrt(1 - 2x + 3x^2 + 2*x^3 + x^4).

Original entry on oeis.org

1, 1, 0, -3, -7, -6, 11, 49, 78, 3, -297, -750, -691, 1271, 5970, 9877, 647, -38640, -100381, -95689, 170394, 827453, 1398933, 131418, -5472241, -14495327, -14186826, 24241947, 121177521, 208360152, 25541493, -807963639, -2175698844, -2179521039

Offset: 0

Seiichi Manyama, Aug 09 2024

sign

Cf. A051286, A108488, A108489.

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-2*x+3*x^2+2*x^3+x^4))

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

n * a(n) = (2*n-1)*a(n-1) - 3*(n-1)*a(n-2) - (2*n-3)*a(n-3) - (n-2)*a(n-4).

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

A376721 Expansion of 1/sqrt((1 - x^3 - x^4)^2 - 4*x^7).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 4, 1, 1, 9, 9, 2, 16, 36, 17, 26, 100, 101, 61, 226, 401, 274, 477, 1227, 1289, 1225, 3186, 4982, 4432, 7841, 16040, 17902, 21457, 45517, 66610, 71327, 123444, 219825, 261945, 354095, 660573, 938598, 1138806, 1909676, 3125553

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

Cf. A051286, A298567, A376722.

Cf. A246883.

my(N=50, x='x+O('x^N)); Vec(1/sqrt((1-x^3-x^4)^2-4*x^7))

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

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

A376722 Expansion of 1/sqrt((1 - x^4 - x^5)^2 - 4*x^9).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 4, 1, 0, 1, 9, 9, 1, 1, 16, 36, 16, 2, 25, 100, 100, 26, 37, 225, 400, 226, 85, 442, 1225, 1226, 505, 833, 3137, 4901, 3217, 2080, 7120, 15878, 15976, 9081, 15696, 44182, 63626, 47125, 41625, 110926, 213688, 217801, 157300, 272251, 630458

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

Cf. A051286, A298567, A376721.

Cf. A246884.

my(N=60, x='x+O('x^N)); Vec(1/sqrt((1-x^4-x^5)^2-4*x^9))

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

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

A081207 Main diagonal of number square A081206.

Original entry on oeis.org

1, 2, 3, 7, 16, 37, 89, 216, 529, 1307, 3248, 8111, 20339, 51176, 129143, 326717, 828374, 2104361, 5354979, 13647682, 34830191, 89000157, 227674188, 583017657, 1494365341, 3833592212, 9842373849, 25287895051, 65016153154, 167264946727

Offset: 0

Paul Barry, Mar 11 2003

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

Cf. A051286, A081206.

Table[Sum[Binomial[Floor[(n+k)/2],k]^2,{k,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 02 2011 *)
CoefficientList[Series[(1+x)/Sqrt[1-2x-x^2-2x^3+x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)

for(n=0,25, print1(sum(k=0,n, (binomial(floor((n+k)/2), k))^2), ", ")) \\ G. C. Greubel, Feb 16 2017

x='x+O('x^25); Vec((1+x)/sqrt(1-2*x-x^2-2*x^3+x^4)) \\ G. C. Greubel, Feb 16 2017

a(n) = Sum_{k=0..n} (binomial(floor((n+k)/2), k))^2.
G.f.: (1+x)/sqrt(1-2x-x^2-2x^3+x^4) - Paul Barry, Jun 04 2005
Conjecture: n*(n-2)*a(n) +(-2*n^2+5*n-1)*a(n-1) +(-n^2+3*n-4)*a(n-2) +(-2*n^2+7*n-4)*a(n-3) +(n-1)*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 12 2012
a(n) ~ (5-sqrt(5)) * ((3+sqrt(5))/2)^n / (2*sqrt(14*sqrt(5)-30) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 04 2014
Equivalently, a(n) ~ 5^(1/4) * phi^(2*n + 1) / (2 * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

A107597 Antidiagonal sums of triangle A107105: a(n) = Sum_{k=0..n} A107105(n-k,k), where A107105(n,k) = C(n,k)*(C(n,k) + 1)/2.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 38, 87, 205, 493, 1203, 2969, 7389, 18504, 46561, 117596, 297883, 756388, 1924484, 4904830, 12519121, 31995286, 81864992, 209681349, 537562018, 1379332297, 3542013533, 9102191107, 23406301490, 60226845008, 155059899921

Offset: 0

Paul D. Hanna, May 22 2005

nonn

Limit a(n+1)/a(n) = (sqrt(5)+3)/2.

Cf. A107105, A051286, A000045.

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

{a(n)= if(n<0, 0, polcoeff( (1/(1-x-x^2) +1/sqrt((1+x+x^2)* (1-3*x+x^2)+ x*O(x^n)))/2, n))} /* Michael Somos, Jul 27 2007 */

a(n) = (A051286(n) + A000045(n+1))/2, where A000045(n+1) = Fibonacci(n+1) and A051286(n) = Whitney number of level n.
G.f.: ( 1/(1-x-x^2) + 1/sqrt( (1+x+x^2)*(1-3*x+x^2) ) )/2. - Michael Somos, Jul 27 2007
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} x^k * C(n,k)*(C(n,k) + 1)/2. - Paul D. Hanna, Aug 13 2014

A182882 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 1. L_n is the set of lattice paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1; an (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 1, 6, 3, 0, 1, 6, 3, 12, 4, 0, 1, 7, 24, 6, 20, 5, 0, 1, 12, 34, 60, 10, 30, 6, 0, 1, 31, 60, 100, 120, 15, 42, 7, 0, 1, 40, 185, 180, 230, 210, 21, 56, 8, 0, 1, 91, 260, 645, 420, 455, 336, 28, 72, 9, 0, 1, 170, 636, 980, 1715, 840, 812, 504, 36, 90, 10, 0, 1

Offset: 0

Emeric Deutsch, Dec 11 2010

Sum of entries in row n is A051286(n).
T(n,0)=A182883(n).
Sum(k*T(n,k), k=0..n)=A182884(n).

			T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them have exactly one h step.
Triangle starts:
1;
0,1;
1,0,1;
2,2,0,1;
1,6,3,0,1;
6,3,12,4,0,1

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A182883, A182884.

G:=1/sqrt(1-2*t*z-2*z^2+t^2*z^2+2*t*z^3+z^4-4*z^3): Gser:=simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form

G.f.: G(t,z) =1/sqrt(1-2tz-2z^2+t^2*z^2+2t*z^3+z^4-4z^3).

A182886 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps. These are paths that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1; a (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 0, 6, 1, 3, 1, 0, 6, 12, 3, 4, 1, 6, 0, 24, 21, 6, 5, 1, 0, 30, 12, 60, 34, 10, 6, 1, 0, 30, 90, 60, 121, 52, 15, 7, 1, 20, 0, 180, 215, 76, 21, 8, 1, 0, 140, 90, 630, 540, 421, 351, 107, 28, 9, 1, 0, 140, 560, 630, 1710, 1176, 846, 539, 146, 36, 10, 1

Offset: 0

Emeric Deutsch, Dec 11 2010

Sum of entries in row n is A051286(n).
T(3n,0) = binomial(2n,n) = A000984(n).
T(n,0) = 0 if n mod 3 > 0.
Sum_{k=0..n} k*T(n,k) = A182887(n).

			T(3,2)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, namely hH and Hh, have exactly two (1,0)-steps.
Triangle starts:
  1;
  0,  1;
  0,  1,  1;
  2,  0,  2,  1;
  0,  6,  1,  3,  1;
  0,  6, 12,  3,  4,  1;

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A182887, A000984.

G:=1/sqrt(1-2*t*z-2*t*z^2+t^2*z^2+2*t^2*z^3+t^2*z^4-4*z^3): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form

G.f.: G(t,z) = 1/sqrt(1 - 2tz - 2tz^2 + t^2*z^2 + 2t^2*z^3 + t^2*z^4 - 4z^3).

A182888 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps at level 0. These are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 8, 7, 6, 4, 0, 1, 17, 20, 12, 8, 5, 0, 1, 38, 44, 36, 18, 10, 6, 0, 1, 89, 104, 82, 56, 25, 12, 7, 0, 1, 206, 253, 204, 132, 80, 33, 14, 8, 0, 1, 485, 604, 513, 344, 195, 108, 42, 16, 9, 0, 1, 1152, 1466, 1262, 891, 530, 272, 140, 52, 18, 10, 0, 1

Offset: 0

Emeric Deutsch, Dec 11 2010

			T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, namely hH and Hh, have exactly two (1,0)-steps at level 0.
Triangle starts:
   1;
   0,   1;
   1,   0,  1;
   2,   2,  0,  1;
   3,   4,  3,  0,  1;
   8,   7,  6,  4,  0,  1;
  17,  20, 12,  8,  5,  0, 1;
  38,  44, 36, 18, 10,  6, 0, 1;
  89, 104, 82, 56, 25, 12, 7, 0, 1;
  ...

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
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.

Row sums give A051286.
Column k=0 gives A182889.
Cf. A182890.

G:=1/(z-t*z+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form

G.f.: G(t,z) = 1/( z-tz+sqrt((1+z+z^2)(1-3z+z^2)) ).

Sum_{k=0..n} k*T(n,k) = A182890(n).

A182893 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps at level 0. The members of L_n are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 4, 1, 3, 1, 4, 8, 6, 3, 4, 1, 12, 12, 18, 9, 6, 5, 1, 24, 36, 30, 32, 14, 10, 6, 1, 54, 84, 78, 64, 51, 22, 15, 7, 1, 130, 184, 204, 152, 120, 77, 34, 21, 8, 1, 300, 452, 462, 416, 280, 205, 113, 51, 28, 9, 1, 706, 1084, 1130, 1000, 770, 492, 328, 163, 74, 36, 10, 1

Offset: 0

Emeric Deutsch, Dec 12 2010

Sum of entries in row n is A051286(n).
T(n,0)=A182894(n).
Sum(k*T(n,k), k=0..n)=A182895(n).

			T(3,2)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, namely hH and Hh, have exactly two (1,0)-steps at level 0.
Triangle starts:
1;
0,1;
0,1,1;
2,0,2,1;
2,4,1,3,1;
4,8,6,3,4,1.

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A182894, A182895.

G:=1/((1-t)*z*(1+z)+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form

G.f.: G(t,z) =1/[(1-t)z(1+z)+sqrt((1+z+z^2)(1-3z+z^2))].

cp's OEIS Frontend

A323769 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375021 Expansion of 1/sqrt(1 - 2*x + 3*x^2 + 2*x^3 + x^4).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A376721 Expansion of 1/sqrt((1 - x^3 - x^4)^2 - 4*x^7).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A376722 Expansion of 1/sqrt((1 - x^4 - x^5)^2 - 4*x^9).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A081207 Main diagonal of number square A081206.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A107597 Antidiagonal sums of triangle A107105: a(n) = Sum_{k=0..n} A107105(n-k,k), where A107105(n,k) = C(n,k)*(C(n,k) + 1)/2.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

PARI

Formula

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Formula

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A375021 Expansion of 1/sqrt(1 - 2x + 3x^2 + 2*x^3 + x^4).