A085362 -id:A085362 - cp's OEIS frontend

A387228 Expansion of sqrt((1-x) / (1-5*x)^5).

1, 12, 103, 764, 5215, 33728, 210021, 1271504, 7532547, 43859460, 251809701, 1428911652, 8028877233, 44734340784, 247433518875, 1359902816880, 7432212863235, 40416897046740, 218812616979845, 1179889937796900, 6339243523221245, 33947223885549040, 181245459484155935

Offset: 0

Seiichi Manyama, Aug 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A387229, A387230.
Cf. A085362, A163869.
Cf. A377199.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- x) / (1-5*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025

CoefficientList[Series[Sqrt[(1-x)/(1-5*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)

my(N=30, x='x+O('x^N)); Vec(sqrt((1-x)/(1-5*x)^5))

n*a(n) = (6*n+6)*a(n-1) - 5*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
a(n) ~ 8 * 5^(n - 1/2) * n^(3/2) / (3*sqrt(Pi)). - Vaclav Kotesovec, Aug 23 2025

A026392 T(n,[ n/2 ]), where T is the array in A026386.

Original entry on oeis.org

1, 2, 4, 8, 17, 34, 75, 150, 339, 678, 1558, 3116, 7247, 14494, 34016, 68032, 160795, 321590, 764388, 1528776, 3650571, 7301142, 17501619, 35003238, 84179877, 168359754, 406020930, 812041860, 1963073865, 3926147730

Offset: 1

Clark Kimberling

nonn

Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 15.

Cf. A026378, A085362.

A026392 := proc(n)
    A026386(n,floor(n/2)) ;
end proc: # R. J. Mathar, Feb 10 2015

Conjecture: (n+1)*a(n) +2*(n-1)*a(n-1) +2*(-3*n+1)*a(n-2) +4*(-3*n+7)*a(n-3) +5*(n-3)*a(n-4) +10*(n-5)*a(n-5)=0. - R. J. Mathar, Feb 10 2015

Offset corrected. R. J. Mathar, Feb 10 2015

A360290 a(n) = Sum_{k=0..floor(n/2)} binomial(n-1-k,k) * binomial(2n-4k,n-2*k).

Original entry on oeis.org

1, 2, 6, 22, 82, 314, 1222, 4814, 19138, 76626, 308550, 1248230, 5069266, 20654602, 84392838, 345659166, 1418769154, 5834283298, 24031706246, 99134911542, 409495076050, 1693539077210, 7011618614342, 29058701620974, 120540377731266, 500443750830962

Offset: 0

Seiichi Manyama, Feb 01 2023

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A085362, A360291, A360292.

Cf. A360185, A360293, A383573.

[&+[Binomial(n-1-k, k) * Binomial(2*n-4*k, n-2*k): k in [0..Floor(n div 2)]]: n in [0..30]]; // Vincenzo Librandi, May 04 2025

Table[Sum[Binomial[n-1-k,k]* Binomial[2*n-4*k, n-2*k],{k,0,Floor[n/2]}],{n,0,35}] (* Vincenzo Librandi, May 04 2025 *)

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

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^2)))

G.f.: 1 / sqrt(1-4*x/(1-x^2)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(n-2)*a(n-2) - 2*(2*n-7)*a(n-3) - (n-4)*a(n-4).
a(n) ~ phi^(3*n) / (5^(1/4) * sqrt(Pi*n/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Feb 02 2023
a(n) = A383573(n) - A383573(n-2). - Seiichi Manyama, May 01 2025

A377200 Expansion of 1/(1 - 4*x/(1-x))^(7/2).

Original entry on oeis.org

1, 14, 140, 1190, 9170, 66122, 454328, 3009050, 19359620, 121664410, 749879508, 4546925922, 27188341530, 160624341990, 939009926520, 5438826037974, 31244200818306, 178173537480330, 1009366349014100, 5684102310204850, 31836106214747590, 177430881586034110

Offset: 0

Seiichi Manyama, Oct 19 2024

nonn

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

Cf. A085362, A377197, A377199.

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x))^(7/2))); // Vincenzo Librandi, May 11 2025

Table[Sum[(-4)^k*Binomial[-7/2,k]*Binomial[n-1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *)

a(n) = sum(k=0, n, (-4)^k*binomial(-7/2, k)*binomial(n-1, n-k));

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (7-5*k/n) * a(k).
a(n) = (2*(3*n+4)*a(n-1) - 5*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n-1,n-k).
a(n) ~ 1024 * 5^(n - 9/2) * n^(5/2) / (3*sqrt(Pi)). - Vaclav Kotesovec, May 03 2025
a(n) = 14*hypergeom([9/2, 1-n], [2], -4) for n > 0. - Stefano Spezia, May 08 2025

A026387 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=0; also a(n) = T(2n,n).

Original entry on oeis.org

2, 8, 34, 150, 678, 3116, 14494, 68032, 321590, 1528776, 7301142, 35003238, 168359754, 812041860, 3926147730, 19022666310, 92338836390, 448968093320, 2186194166950, 10659569748370, 52037098259090, 254308709196660

Offset: 0

Clark Kimberling

László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

Essentially the same as A085362.

Cf. A026378.

a(n) = A085362(n+1), n >= 0. - Hartmut F. W. Hoft, Jul 07 2024

A361817 Expansion of 1/sqrt(1 - 4x(1-x)^4).

Original entry on oeis.org

1, 2, -2, -16, -10, 118, 304, -500, -3754, -2488, 30866, 83716, -135568, -1080972, -792876, 9090484, 25788118, -39325156, -335074520, -271779024, 2820643842, 8348113120, -11788972644, -107836934448, -96107852032, 900943403012, 2778574561276, -3596374190416

Offset: 0

Seiichi Manyama, Mar 25 2023

sign

Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361815, A361816.

Cf. A361813.

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x)^4))

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

n*a(n) = 2 * ( (2*n-1)*a(n-1) - 4*(2*n-2)*a(n-2) + 6*(2*n-3)*a(n-3) - 4*(2*n-4)*a(n-4) + (2*n-5)*a(n-5) ) for n > 4.

A372109 G.f. A(x) satisfies A(x) = ( (1 - xA(x))/(1 - 5x*A(x)) )^(1/2).

Original entry on oeis.org

1, 2, 12, 90, 758, 6850, 64904, 636250, 6399120, 65661250, 684665828, 7233956250, 77278356246, 833291781250, 9057750917944, 99144375156250, 1091857567068742, 12089416175781250, 134501879883249300, 1502857085910156250, 16857310306553767026

Offset: 0

Seiichi Manyama, Apr 19 2024

nonn

Index entries for reversions of series

Cf. A001003, A372110.

Cf. A085362, A372018, A372090, A372104, A378551.

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

a(n) = (1/(n+1)) * Sum_{k=0..n} 4^k * binomial(n/2+k-1/2,k) * binomial(n-1,n-k).
From Seiichi Manyama, Nov 30 2024: (Start)
G.f.: exp( Sum_{k>=1} A378551(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - 4*x/(1-x))^((n+1)/2).
G.f.: (1/x) * Series_Reversion( x*(1 - 4*x/(1-x))^(1/2) ). (End)

A101894 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at odd height.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 8, 3, 1, 36, 34, 15, 4, 1, 137, 150, 77, 24, 5, 1, 543, 678, 399, 144, 35, 6, 1, 2219, 3116, 2073, 854, 240, 48, 7, 1, 9285, 14494, 10769, 4996, 1600, 370, 63, 8, 1, 39587, 68032, 55875, 28852, 10387, 2736, 539, 80, 9, 1, 171369, 321590, 289431

Offset: 0

Emeric Deutsch, Dec 20 2004

A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318). Row sums are the large Schroeder numbers (A006318). Column 0 yields A002212. Column 1 yields A085362.

			T(3,2)=3 because we have H(UD)(UD), (UD)(UD)H and (UD)H(UD), the peaks at aodd height being shown between parentheses.
Triangle begins:
1;
1,1;
3,2,1;
10,8,3,1;
36,34,15,4,1;

Cf. A006318, A002212, A085362, A101895.

G := 1/2/(-z+t*z^2)*(-1+t*z+z-t*z^2+sqrt(1-2*t*z-6*z+8*t*z^2+t^2*z^2-2*t^2*z^3+5*z^2-6*t*z^3+t^2*z^4)): Gser:=simplify(series(G,z=0,13)):P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od;

G.f.=G=G(t, z) satisfies z(1-tz)G^2-(1-z)(1-tz)G+1-z=0.

A128749 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k ascents of length 1.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 14, 12, 9, 0, 1, 44, 53, 25, 14, 0, 1, 150, 196, 132, 44, 20, 0, 1, 520, 777, 555, 269, 70, 27, 0, 1, 1850, 3064, 2486, 1260, 485, 104, 35, 0, 1, 6696, 12233, 10902, 6264, 2496, 804, 147, 44, 0, 1, 24602, 49096, 47955, 30108, 13600

Offset: 0

Emeric Deutsch, Mar 31 2007

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps. An ascent in a path is a maximal sequence of consecutive U steps.

Row sums yield A002212.

			T(3,1)=5 because we have (U)DUUDD, (U)DUUDL, UUDD(U)D, UUD(U)DD and UUD(U)DL (the ascents of length 1 are shown between parentheses).
Triangle starts:
   1;
   0,  1;
   2,  0,  1;
   4,  5,  0,  1;
  14, 12,  9,  0,  1;
  44, 53, 25, 14,  0,  1;

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Cf. A002212, A085362, A128750.

eq:=z*(1+z-t*z)*G^2-(1-t*z+t*z^2-z^2)*G+1-z=0: G:=RootOf(eq,G): 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,j),j=0..n) od; # yields sequence in triangular form

T(n,0) = A128750(n).
Sum_{k=0..n} k*T(n,k) = A085362(n-1).
G.f.: G = G(t,z) satisfies z(1 + z - tz)G^2 - (1 - tz + tz^2 - z^2)G + 1 - z = 0.

A372104 G.f. A(x) satisfies A(x) = 1/( 1 - 4xA(x)/(1-x) )^(1/2).

Original entry on oeis.org

1, 2, 12, 86, 686, 5858, 52404, 484814, 4600652, 44534386, 438034928, 4365350062, 43983695242, 447305878226, 4585518132768, 47335424695142, 491615988964766, 5133343692822146, 53858312462193328, 567501135052136702, 6002857276349252630

Offset: 0

Seiichi Manyama, Apr 18 2024

nonn

Cf. A085362, A372086.

a(n) = sum(k=0, n, 4^k*binomial(3*k/2-1/2, k)*binomial(n-1, n-k)/(k+1));

a(n) = Sum_{k=0..n} 4^k * binomial(3*k/2-1/2,k) * binomial(n-1,n-k)/(k+1).

cp's OEIS Frontend

A387228 Expansion of sqrt((1-x) / (1-5*x)^5).

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A026392 T(n,[ n/2 ]), where T is the array in A026386.

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A360290 a(n) = Sum_{k=0..floor(n/2)} binomial(n-1-k,k) * binomial(2*n-4*k,n-2*k).

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A377200 Expansion of 1/(1 - 4*x/(1-x))^(7/2).

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A026387 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=0; also a(n) = T(2n,n).

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A361817 Expansion of 1/sqrt(1 - 4*x*(1-x)^4).

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A372109 G.f. A(x) satisfies A(x) = ( (1 - x*A(x))/(1 - 5*x*A(x)) )^(1/2).

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A101894 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at odd height.

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A128749 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k ascents of length 1.

A360290 a(n) = Sum_{k=0..floor(n/2)} binomial(n-1-k,k) * binomial(2n-4k,n-2*k).

A361817 Expansion of 1/sqrt(1 - 4x(1-x)^4).

A372109 G.f. A(x) satisfies A(x) = ( (1 - xA(x))/(1 - 5x*A(x)) )^(1/2).

A372104 G.f. A(x) satisfies A(x) = 1/( 1 - 4xA(x)/(1-x) )^(1/2).