Eric Werley - cp's OEIS frontend

A261819 Encoded symmetrical antidiagonal square binary matrices with either 1 or 2 ones.

1, 6, 16, 40, 384, 576, 4096, 10240, 17408, 393216, 589824, 1081344, 16777216, 41943040, 71303168, 136314880, 6442450944, 9663676416, 17716740096, 34628173824, 1099511627776

Offset: 0

Eric Werley, Sep 24 2015

nonn

We encode square matrices that have zeros everywhere except the antidiagonal where the antidiagonal is symmetric with either 1 or 2 ones in it. We do this by reading off digits antidiagonally to get a binary number and then convert the number to a base 10 number.

			The 3 X 3 matrix
0 0 0
0 1 0
0 0 0
gives 000010000. Writing this as a base 10 number gives a(2)=16.
The 4 X 4 matrix
0 0 0 0
0 0 1 0
0 1 0 0
0 0 0 0
gives 0000000110000000. Writing this as a base 10 number gives a(4)=384.
The 5 X 5 matrix
0 0 0 0 0
0 0 0 1 0
0 0 0 0 0
0 1 0 0 0
0 0 0 0 0
gives 0000000000010100000000000. Writing this as a base 10 number gives a(7)=10240.

a(n) = A261195(2^n).

a(n) = 2^(A000217(floor(sqrt(4*n + 1)) - 1)) * (((A262769(floor(n/2)) * 2^((floor(sqrt(4*n + 1)) - 2*A002260(+1))/2)) * (1+(-1)^(floor(sqrt(4*n + 1))))/2) + ((A262777(floor(n/2)) * 2^((floor(sqrt(4*n + 1)) - A158405(+1))/2)) * (1-(-1)^(floor(sqrt(4*n + 1))))/2)).

A239767 Degrees of polynomial on the fermionic side of the finite generalization of identity 46 from Slater's List.

Original entry on oeis.org

0, 1, 6, 11, 22, 31, 48, 61, 84, 101, 130, 151, 186, 211, 252, 281, 328, 361, 414, 451, 510, 551, 616, 661, 732, 781, 858, 911, 994, 1051, 1140, 1201, 1296, 1361, 1462, 1531, 1638, 1711, 1824, 1901, 2020, 2101, 2226, 2311, 2442, 2531, 2668, 2761, 2904, 3001

Offset: 0

Eric Werley, Mar 26 2014

A "Rogers-Ramanujan-Slater" type identity is an identity containing a variable q which equates an infinite product with an infinite series. A finite generalization of such an identity consists of two sequences of polynomials, such that corresponding terms in each sequence are equal and one sequence tends to the infinite sum and the other sequence tends to the infinite product. [From AMS Abstracts 2008 Eric Werley by Michael Somos, Mar 27 2014]

In statistical mechanics, the fermionic side of a Rogers-Ramanujan type identity is the infinite series side of the identity and the bosonic side is the infinite product side of the identity.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
George E. Andrews, The hard-hexagon model and Rogers-Ramanujan type identities, Proc. Nat. Acad. Sci. U.S.A., 78(1981), 5290-5292.
L. J. Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., 54(1952), 147-167.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

[(1/8)*(10*n^2+2*(1+(-1)^n)*n-(1-(-1)^n)): n in [0..50]]; // Vincenzo Librandi, Mar 29 2014

A239767:=n->(10*n^2 + 2*n*(1+(-1)^n) - (1-(-1)^n))/8; seq(A239767(n), n=0..100); # Wesley Ivan Hurt, Mar 27 2014

Table[(10 n^2 + 2 n (1 + (-1)^n) - (1 - (-1)^n))/8, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 27 2014 *)
CoefficientList[Series[- x (x^3 + 3 x^2 + 5 x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 29 2014 *)

concat(0, Vec(-x*(x^3+3*x^2+5*x+1)/((x-1)^3*(x+1)^2) + O(x^100))) \\ Colin Barker, Mar 26 2014

a(n) = (1/8)*(10*n^2 + 2*(1+(-1)^n)*n - (1-(-1)^n)).
From Colin Barker, Mar 26 2014: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(x^3+3*x^2+5*x+1) / ((x-1)^3*(x+1)^2). (End)

More terms from Colin Barker, Mar 26 2014

A192136 a(n) = (5n^2 - 3n + 2)/2.

Original entry on oeis.org

1, 2, 8, 19, 35, 56, 82, 113, 149, 190, 236, 287, 343, 404, 470, 541, 617, 698, 784, 875, 971, 1072, 1178, 1289, 1405, 1526, 1652, 1783, 1919, 2060, 2206, 2357, 2513, 2674, 2840, 3011, 3187, 3368, 3554, 3745, 3941, 4142, 4348, 4559, 4775, 4996, 5222, 5453, 5689

Offset: 0

Eric Werley, Jun 24 2011

Binomial transform of [1, 1, 5, 0, 0, 0, 0, 0, ...]. - Johannes W. Meijer, Jul 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A000217, A002522, A006137, A130883, A140063, A140064, A143689.

A192136:=func< n | (5*n^2-3*n+2)/2 >; [ A192136(n): n in [0..50] ]; // Klaus Brockhaus, Jun 27 2011

Table[(5n^2-3n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,2,8},50] (* Harvey P. Dale, Aug 08 2016 *)

a(n)=n*(5*n-3)/2+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (5*n^2 - 3*n + 2)/2.
a(n) = 2*a(n-1) - a(n-2) + 5.
a(n) = a(n-1) + 5*n - 4.
a(n) = 5*binomial(n+2,2) - 9*n - 4.
a(n) = A000217(n+1) - A000217(n) + 5*A000217(n-1); triangular numbers. - Johannes W. Meijer, Jul 07 2011
O.g.f.: (1-x+5*x^2)/(1-x)^3.
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: exp(x)*(2 + 2*x + 5*x^2)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A192364 Number of lattice paths from (0,0) to (n,n) using steps (0,1),(0,2),(1,0),(2,0),(1,1).

Original entry on oeis.org

1, 3, 21, 157, 1239, 10047, 82951, 693603, 5854581, 49778997, 425712429, 3657968097, 31555053921, 273109567797, 2370474720369, 20625186298269, 179841473895447, 1571088267426447, 13747953837604959, 120482775658910763, 1057293764707074027, 9289536349244758791, 81709329486947791419

Offset: 0

Eric Werley, Jun 29 2011

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A091533.

FullSimplify[CoefficientList[Series[(3-6*x+Sqrt[-1+4*x*(9*x-11)+4*Sqrt[1-x]*Sqrt[5+4*x]*Sqrt[9*x-1]])/(Sqrt[10+8*x]*Sqrt[(1-x)*(1-9*x)]*(4*x*(9*x-11)-1+4*Sqrt[1-x]*Sqrt[5+4*x]*Sqrt[9*x-1])^(1/4)), {x, 0, 10}], x]]

/* same as in A092566 but use */
steps=[[0,1], [0,2], [1,0], [2,0], [1,1]];
/* Joerg Arndt, Jun 30 2011 */

From Vaclav Kotesovec, Oct 24 2012: (Start)
G.f.: (3 - 6*x + sqrt(-1 + 4*x*(9*x-11) + 4*sqrt(1-x)*sqrt(5+4*x)*sqrt(9*x-1))) / (sqrt(10+8*x)*sqrt((1-x)*(1-9*x))*(4*x*(9*x-11)-1+4*sqrt(1-x)*sqrt(5+4*x)*sqrt(9*x-1))^(1/4))
D-finite with recurrence: 15*(n-1)*n*a(n) = (n-1)*(133*n-54)*a(n-1) + (31*n^2 - 177*n + 224)*a(n-2) - (113*n^2 - 295*n + 144)*a(n-3) - 18*(n-3)*(2*n-5)*a(n-4)
a(n) ~ 3^(2*n+3/2)/(2*sqrt(14*Pi*n))
(End)
a(n) = A091533(2*n,n) for n >= 0. - Paul D. Hanna, Dec 11 2018
a(n) = [x^n*y^n] 1/(1 - x - y - x^2 - x*y - y^2) for n >= 0. - Paul D. Hanna, Dec 11 2018

Terms > 425712429 by Joerg Arndt, Jun 30 2011

A192365 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(2,0),(0,1),(0,2),(1,1),(2,2).

Original entry on oeis.org

1, 3, 22, 165, 1327, 10950, 92045, 783579, 6733966, 58294401, 507579829, 4440544722, 39000863629, 343677908223, 3037104558574, 26904952725061, 238854984979423, 2124492829796598, 18927927904130617, 168888613467092895, 1508973226894216106, 13498652154574126523, 120886709687492946083

Offset: 0

Eric Werley, Jun 29 2011

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000

p4 := x^4+6*x^3+7*x^2-10*x+1;
ogf := sqrt( ((2*x^2+6*x-3)/p4 - 2/sqrt(p4))/(4*x^2-4*x-5) );
series(ogf, x=0, 30);  # Mark van Hoeij, Apr 16 2013
# second Maple program:
b:= proc(x, y) option remember; `if`(min(x, y)<0, 0,
      `if`(max(x, y)=0, 1, add(b(x, y-j)+
         b(x-j, y)+b(x-j, y-j), j=1..2)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2017

b[x_, y_] := b[x, y] = If[Min[x, y] < 0, 0, If[Max[x, y] == 0, 1, Sum[b[x, y - j] + b[x - j, y] + b[x - j, y - j], {j, 1, 2}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 23 2017, after Alois P. Heinz *)

/* same as in A092566 but use */
steps=[[0,1], [0,2], [1,0], [2,0], [1,1], [2,2]];
/* Joerg Arndt, Jun 30 2011 */

G.f.: sqrt( ((2*x^2+6*x-3)/p4 - 2/sqrt(p4))/(4*x^2-4*x-5) ) where p4 = x^4+6*x^3+7*x^2-10*x+1. - Mark van Hoeij, Apr 16 2013

Terms > 507579829 from Joerg Arndt, Jun 30 2011

A192369 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (0,2), (1,0), (2,0), and (2,2).

Original entry on oeis.org

1, 2, 15, 90, 617, 4248, 29945, 213404, 1535661, 11129314, 81123369, 594092166, 4367701295, 32216566492, 238301617605, 1766979857196, 13129849298327, 97746629874786, 728897653778335, 5443488765350770, 40706993579981847, 304779612155116444, 2284440756129389775, 17139937071103287600

Offset: 0

Eric Werley, Jun 29 2011

nonn

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

p4 := (x-1)*(x^3+5*x^2+7*x-1);
ogf := sqrt(((2*x^2+4*x-3)/p4-2/sqrt(p4))/(4*x^2-8*x-5));
series(ogf, x=0, 30); # Mark van Hoeij, Apr 16 2013

/* same as in A092566 but use */
steps=[[0,1], [0,2], [1,0], [2,0], [2,2]];
/* Joerg Arndt, Jun 30 2011 */

G.f. is a nested square root, see Maple program. - Mark van Hoeij, Apr 16 2013

Terms > 81123369 from Joerg Arndt, Jun 30 2011

A175900 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(1,1)|0

Original entry on oeis.org

1, 2, 9, 57, 411, 3150, 25274, 209719, 1784909, 15495058, 136672624, 1221370092, 11034687854, 100623850938, 924917387599, 8560734224711, 79717622732369, 746322905220237, 7020573752016609, 66324819580927731, 629004053328092981

Offset: 0

Eric Werley, Dec 05 2010

nonn

Cf. A006318, A175895, A175896

A175935 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|0

Original entry on oeis.org

1, 2, 10, 55, 351, 2401, 17248, 128221, 978082, 7612155, 60204488, 482481220, 3909460725, 31974923487, 263623879118, 2188682538746, 18282238300443, 153537981720402, 1295640515428649, 10980400434511117, 93418283866708579

Offset: 0

Eric Werley, Dec 06 2010

nonn

Cf. A175934, A175936, A175937, A175939

A175937 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|0

Original entry on oeis.org

1, 2, 10, 62, 448, 3464, 28111, 236022, 2033145, 17867442, 159558635, 1443747386, 13207922431, 121962046864, 1135246916024, 10640772522150, 100346005711723, 951400275042466, 9063703952844960, 86718277215053218, 832901296331740527

Offset: 0

Eric Werley, Dec 06 2010

nonn

Cf. A175934, A175935, A175936, A175939

A175912 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(1,1)|k>0} which never go above the line y=x.

Original entry on oeis.org

1, 2, 9, 57, 411, 3181, 25803, 216486, 1863139, 16356925, 145914573, 1318844414, 12051758083, 111159508991, 1033505202643, 9675905948106, 91140492185703, 863107104436546, 8212873185281571, 78484928498979435, 752928813642151089

Offset: 0

Eric Werley, Dec 05 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - From _N. J. A. Sloane_, Dec 27 2012
Joseph P. S. Kung, Anna de Mier, Rook and queen paths with boundaries, arXiv:1109.1806.

Cf. A006318, A175895, A175896, A175900.

Flatten[{1,RecurrenceTable[{(n+1)*a[n]-10*(n+2)*a[n+1]+(34*n+96)*a[n+2]-6*(8*n+29)*a[n+3]+5*(5*n+23)*a[n+4]-2*(n+6)*a[n+5]==0, a[1]==2, a[2]==9, a[3]==57, a[4]==411, a[5]==3181},a,{n,20}]}] (* Vaclav Kotesovec, Sep 07 2012 *)

Asymptotic: a(n) ~ b*c^n/n^(3/2), where c = 10.33185141266662366... is the root of the equation c^3-11*c^2+7*c-1=0 and b = sqrt(13*c-5-c^2)*(2*c^2+9*c-2)/(2*c^3*sqrt(Pi)) = 0.36996178... - Vaclav Kotesovec, Dec 25 2013

G.f. (from reference): (1+2*x-x^2 - sqrt((x-1)*(x^3-7*x^2+11*x-1)))/(2*x*(x-2)^2). - Vaclav Kotesovec, Dec 25 2013

Minor edits Vaclav Kotesovec, Mar 31 2014

cp's OEIS Frontend

User: Eric Werley

A261819 Encoded symmetrical antidiagonal square binary matrices with either 1 or 2 ones.

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A239767 Degrees of polynomial on the fermionic side of the finite generalization of identity 46 from Slater's List.

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A192136 a(n) = (5*n^2 - 3*n + 2)/2.

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A192364 Number of lattice paths from (0,0) to (n,n) using steps (0,1),(0,2),(1,0),(2,0),(1,1).

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A192365 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(2,0),(0,1),(0,2),(1,1),(2,2).

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Maple

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A192369 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (0,2), (1,0), (2,0), and (2,2).

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Maple

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Extensions

A175900 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(1,1)|0

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A175935 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|0

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A175937 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|0

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A192136 a(n) = (5n^2 - 3n + 2)/2.