A059317 -id:A059317 - cp's OEIS frontend

A059345 Central column of Pascal's "rhombus" (actually a triangle) A059317.

1, 1, 4, 9, 29, 82, 255, 773, 2410, 7499, 23575, 74298, 235325, 747407, 2381126, 7603433, 24332595, 78013192, 250540055, 805803691, 2595158718, 8368026845, 27012184877, 87283372610, 282294378071, 913775677281, 2960160734818

Offset: 0

N. J. A. Sloane, Jan 27 2001

Number of paths in the right half-plane from (0,0) to (n,0) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(3)=9 because we have hhh, hH, Hh, hUD, hDU, UhD, DhU, UDh and DUh. The number of such paths restricted to the first quadrant is given in A128720. - Emeric Deutsch, Sep 03 2007
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), (2,2). - Joerg Arndt, Jun 30 2011
Other two columns of the triangle in A059317 are given in A106053 and A106050. - Emeric Deutsch, Sep 03 2007

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

T. D. Noe, Table of n, a(n) for n=0..200
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 231-236.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv:1504.04404 [math.CO], 2015.

Cf. A128720, A106050, A106053.

Cf. A181545. - Paul D. Hanna, Oct 29 2010

r:=proc(i,j) if i=0 then 0 elif i=1 and abs(j)>0 then 0 elif i=1 and j=0 then 1 elif i>=1 then r(i-1,j)+r(i-1,j-1)+r(i-1,j+1)+r(i-2,j) else 0 fi end: seq(r(i,0),i=1..12); # very slow; Emeric Deutsch, Jun 06 2004
G:=1/sqrt((1+z-z^2)*(1-3*z-z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z, n),n=0..27); # Emeric Deutsch, Sep 03 2007
a[0]:=1: a[1]:=1: a[2]:=4: a[3]:=9: for n from 3 to 26 do a[n+1]:=((2*n+1)*a[n]+5*n*a[n-1]-(2*n-1)*a[n-2]-(n-1)*a[n-3])/(n+1) end do: seq(a[n],n=0..27); # Emeric Deutsch, Sep 03 2007

CoefficientList[Series[1/Sqrt[(1+x-x^2)(1-3x-x^2)],{x,0,40}],x] (* Harvey P. Dale, Jun 04 2011 *)
a[n_] := Sum[Binomial[n-k, k]*Hypergeometric2F1[(2*k-n)/2, (2*k-n+1)/2, 1, 4], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 26 2015 *)

{a(n)=polcoeff(sum(m=0,n,x^(2*m)/(1-x-x^2+x*O(x^n))^(2*m+1)*(2*m)!/(m!)^2),n)} \\ Paul D. Hanna, Oct 29 2010

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

G.f.: 1/sqrt((1+z-z^2)*(1-3*z-z^2)). - Emeric Deutsch, Sep 03 2007
D-finite with recurrence: (n+1)*a(n+1)=(2*n+1)*a(n)+5*n*a(n-1)-(2*n-1)*a(n-2)-(n-1)*a(n-3). - Emeric Deutsch, Sep 03 2007
a(n) = sum{k=0..floor(n/2), C(n-k,k)*A002426(n-2k)}. - Paul Barry, Nov 29 2008
G.f.: A(x) = Sum_{n>=0} (2*n)!/(n!)^2 * x^(2n)/(1-x-x^2)^(2n+1). - Paul D. Hanna, Oct 29 2010
a(n) ~ sqrt((3+11/sqrt(13))/8) * ((3+sqrt(13))/2)^n/sqrt(Pi*n). - Vaclav Kotesovec, Aug 11 2013

More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001

A106053 Next-to-central column of triangle in A059317.

Original entry on oeis.org

0, 0, 1, 2, 8, 22, 72, 218, 691, 2158, 6833, 21612, 68726, 218892, 699197, 2237450, 7174018, 23038582, 74097134, 238625222, 769407486, 2483532218, 8024499657, 25951580444, 83999410292, 272098963300, 882045339733, 2861184745710, 9286923094550, 30161343633746

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

Number of h steps in all paths in the first quadrant from (0,0) to (n-1,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because in the 6 (=A128720(3)) paths hhh, hH, Hh, hUD, UhD and UDh we have altogether 8 h-steps. a(n) = Sum_{k=0..n-1} k*A132277(n-1,k). - Emeric Deutsch, Sep 03 2007

Number of paths in the right half-plane from (0,0) to (n-1,1) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because we have hhU, HU, hUh, Uhh, UH, DUU, UDU and UUD. Number of h-steps in all paths in the first quadrant from (0,0) to (n-1,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because in the 6 (=A128720(3)) paths from (0,0) to (3,0), namely, hhh, hH, Hh, hUD, UhD and UDh, we have altogether 8 h-steps. a(n) = Sum_{k=0..n-1} k*A132277(n-1,k). - Emeric Deutsch, Sep 03 2007

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
José L. Ramírez, The Pascal Rhombus and the Generalized Grand Motzkin Paths, arXiv:1511.04577 [math.CO], 2015.

Cf. A059317, A128720, A132277.

g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/sqrt((1+z-z^2)*(1-3*z-z^2)): gser:=series(g,z=0,33); seq(coeff(gser,z,n),n=0..29); # Emeric Deutsch, Sep 03 2007
g:=((1-z-z^2)*1/2)/sqrt((1+z-z^2)*(1-3*z-z^2))-1/2: gser:=series(g,z=0,33): seq(coeff(gser,z,n),n=0..30); # Emeric Deutsch, Sep 03 2007

t[0, 0] = t[1, 0] = t[1, 1] = t[1, 2] = 1;
t[n_ /; n >= 0, k_ /; k >= 0] /; k <= 2n := t[n, k] = t[n-1, k] + t[n-1, k-1] + t[n-1, k-2] + t[n-2, k-2];
t[n_, k_] /; n<0 || k<0 || k>2n = 0;
a[n_] := t[n-1, n-2];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Aug 07 2018 *)

G.f.: (1 - z - z^2 - sqrt((1+z-z^2)*(1-3z-z^2)))/(2*sqrt((1+z-z^2)*(1-3z-z^2))). - Emeric Deutsch, Sep 03 2007

G.f.: (1-z-z^2)/(2*sqrt((1+z-z^2)*(1-3z-z^2))) - 1/2. - Emeric Deutsch, Sep 03 2007

A106050 Column two-from-center of triangle A059317.

Original entry on oeis.org

0, 0, 0, 1, 3, 13, 42, 146, 476, 1574, 5122, 16706, 54256, 176254, 571954, 1856245, 6023681, 19551939, 63476314, 206145075, 669695819, 2176401235, 7075521724, 23011145314, 74864599954, 243652588070, 793264765396, 2583532274289, 8416929889967, 27430452311513

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

Number of paths in the right-half-plane from (0,0) to (n-1,2) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=3 because we have hUU, UhU and UUh. - Emeric Deutsch, Sep 03 2007

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
José L. Ramírez, The Pascal Rhombus and the Generalized Grand Motzkin Paths, arXiv:1511.04577 [math.CO], 2015.

Cf. A059317, A059345, A106053.

g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/z^2: gser:=series(z^3*g^2/sqrt((1+z-z^2)*(1-3*z-z^2)),z=0,32): seq(coeff(gser,z,n),n=0..30); # Emeric Deutsch, Sep 03 2007

t[0, 0] = t[1, 0] = t[1, 1] = t[1, 2] = 1;
t[n_ /; n >= 0, k_ /; k >= 0] /; k <= 2 n := t[n, k] = t[n-1, k] + t[n-1, k-1] + t[n-1, k-2] + t[n-2, k-2];
t[n_, k_] /; n<0 || k<0 || k>2n = 0;
a[n_] := t[n-1, n-3];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Aug 07 2018 *)

G.f.: z^3*g^2/sqrt((1+z-z^2)(1-3z-z^2)), where g=1+zg+z^2*g+z^2*g^2=[1-z-z^2-sqrt((1+z-z^2)(1-3z--z^2))]/(2z^2). - Emeric Deutsch, Sep 03 2007

A106058 4th diagonal of triangle in A059317.

Original entry on oeis.org

0, 0, 0, 2, 9, 22, 42, 70, 107, 154, 212, 282, 365, 462, 574, 702, 847, 1010, 1192, 1394, 1617, 1862, 2130, 2422, 2739, 3082, 3452, 3850, 4277, 4734, 5222, 5742, 6295, 6882, 7504, 8162, 8857, 9590, 10362, 11174, 12027, 12922, 13860, 14842, 15869, 16942, 18062

Offset: 0

N. J. A. Sloane, May 28 2005

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A007290, A059317.

Join[{0,0}, LinearRecurrence[{4, -6, 4, -1}, {0, 2, 9, 22}, 45]] (* Georg Fischer, Dec 10 2019 *)

a(n)=if(n>2,(n-2)*(n^2 + 8*n - 21)/6,0) \\ Charles R Greathouse IV, Oct 18 2022

For n>1, a(n) = (1/6)*(n-2)*(n^2 + 8n - 21).
From R. J. Mathar, Feb 06 2010: (Start)
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) for n >= 6.
G.f.: -x^3*(-2-x+2*x^2)/(x-1)^4. (End)
E.g.f.: exp(x)*(42 - 30*x + 9*x^2 + x^3)/6 - 7 - 2*x. - Stefano Spezia, Aug 06 2025

A267192 Column 3 of triangle in A059317 (the Pascal "Rhombus").

Original entry on oeis.org

0, 0, 0, 1, 4, 19, 70, 261, 914, 3177, 10816, 36566, 122552, 408840, 1358032, 4497995, 14862112, 49019688, 161449208, 531152855, 1745892452, 5734722698, 18826352472, 61777432510, 202648614072, 664569581090, 2178948104572, 7143067052707, 23413795288008

Offset: 0

N. J. A. Sloane, Jan 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
José L. Ramírez, The Pascal Rhombus and the Generalized Grand Motzkin Paths, arXiv:1511.04577 [math.CO], 2015.

Cf. A059317, A106050, A106053.

T:= proc(n, k) option remember; `if`(min(n, k)<0, 0,
      `if`(k=0, 1, T(n-1, k)+T(n-1, k-1)+T(n-1, k-2)+T(n-2, k-2)))
    end:
a:= n-> T(n, n-3):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 24 2016

T[n_, k_] := T[n, k] = If[Min[n, k]<0, 0, If[k == 0, 1, T[n-1, k] + T[n-1, k-1] + T[n-1, k-2] + T[n-2, k-2]]];
a[n_] := T[n, n-3];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 23 2017, after Alois P. Heinz *)

Conjecture: +(n-2)*(n-3)*(n+3)*a(n) -n*(2*n-1)*(n-2)*a(n-1) -(n-1)*(5*n^2-10*n+18)*a(n-2) +n*(2*n-3)*(n-2)*a(n-3) +n*(n+1)*(n-5)*a(n-4)=0. - R. J. Mathar, Jul 23 2017

More terms from Alois P. Heinz, Jan 24 2016

A106113 5th diagonal of triangle in A059317.

Original entry on oeis.org

0, 0, 0, 1, 8, 29, 72, 146, 261, 428, 659, 967, 1366, 1871, 2498, 3264, 4187, 5286, 6581, 8093, 9844, 11857, 14156, 16766, 19713, 23024, 26727, 30851, 35426, 40483, 46054, 52172, 58871, 66186, 74153, 82809, 92192, 102341, 113296, 125098, 137789, 151412

Offset: 0

N. J. A. Sloane, May 28 2005

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

a(n)=if(n>2,n^4 + 14*n^3 - 97*n^2 + 154*n - 24,0)/24 \\ Charles R Greathouse IV, Oct 21 2022

For n>2, a(n) = (1/24) [n^4 + 14n^3 - 97n^2 + 154n - 24 ].
From Chai Wah Wu, Mar 11 2021: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 7.
G.f.: x^3*(-x^4 + 3*x^3 + x^2 - 3*x - 1)/(x - 1)^5. (End)

A106150 6th diagonal of triangle in A059317.

Original entry on oeis.org

0, 0, 0, 0, 3, 22, 82, 218, 476, 914, 1603, 2628, 4089, 6102, 8800, 12334, 16874, 22610, 29753, 38536, 49215, 62070, 77406, 95554, 116872, 141746, 170591, 203852, 242005, 285558, 335052, 391062, 454198, 525106, 604469, 693008, 791483, 900694, 1021482

Offset: 0

N. J. A. Sloane, May 28 2005

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

a(n)=if(n>3,(n-3)*(n^4+28*n^3-71*n^2-478*n+1360)/120,0) \\ Charles R Greathouse IV, Oct 21 2022

For n>2, a(n) = (1/120) (n-3)(n^4+28n^3-71n^2-478n+1360).
From Chai Wah Wu, Mar 11 2021: (Start)
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 8.
G.f.: x^4*(-3*x^2 + x + 3)*(-x^2 + x + 1)/(x - 1)^6. (End)

A106173 7th diagonal of triangle in A059317.

Original entry on oeis.org

0, 0, 0, 0, 1, 13, 72, 255, 691, 1574, 3177, 5867, 10121, 16543, 25882, 39051, 57147, 81472, 113555, 155175, 208385, 275537, 359308, 462727, 589203, 742554, 927037, 1147379, 1408809, 1717091, 2078558, 2500147, 2989435, 3554676, 4204839, 4949647, 5799617

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

a(n) is a 6th degree polynomial in n.

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,1,13,72,255,691,1574,3177},40] (* Harvey P. Dale, Jun 26 2022 *)

From Chai Wah Wu, Feb 28 2018: (Start)
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 10.
G.f.: x^4*(x^6 - 6*x^5 + 2*x^4 + 11*x^3 - 2*x^2 - 6*x - 1)/(x - 1)^7. (End)

A160905 Right hand side of Pascal rhombus A059317.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 9, 8, 3, 1, 29, 22, 13, 4, 1, 82, 72, 42, 19, 5, 1, 255, 218, 146, 70, 26, 6, 1, 773, 691, 476, 261, 107, 34, 7, 1, 2410, 2158, 1574, 914, 428, 154, 43, 8, 1, 7499, 6833, 5122, 3177, 1603, 659, 212, 53, 9, 1, 23575, 21612, 16706, 10816, 5867, 2628, 967

Offset: 0

Paul Barry, May 29 2009

Riordan array (1/sqrt((1+x-x^2)*(1-3*x-x^2)), (1-x-x^2-sqrt((1+x-x^2)*(1-3*x-x^2)))/(2*x)). Can be factored as
(1/(1-x-x^2), x/(1-x-x^2))*(1/sqrt(1-4x^2),xc(x^2)) = (1/(1-x^2),x/(1-x^2))*(1/(1-x),x/(1-x))*(1/sqrt(1-4x^2),xc(x^2))
and (1/(1-x^2),x/(1-x^2))*(1/sqrt(1-2x-3x^2),(1-x-sqrt(1-2x-3x^2))/(2x)).
Here, c(x) is the g.f. of the Catalan numbers A000108.
A160905 = A037027*A108044 = abs(A049310)*A007318*A108044 = abs(A049310)*A094531.

			Triangle begins:
    1;
    1,   1;
    4,   2,   1;
    9,   8,   3,  1;
   29,  22,  13,  4,  1;
   82,  72,  42, 19,  5, 1;
  255, 218, 146, 70, 26, 6, 1;
  ...

Left column gives A059345.

Cf. A059317.

Number triangle T(n,k) = Sum_{i=0..n} (Sum_{j=0..n} C((n+j)/2,j)*C(j,i)*(1+(-1)^(n-j))/2)*C(i,(i-k)/2)*(1+(-1)^(i-k))/2;

T(n,k) = Sum_{j=0..n} C((n+j)/2,j)*((1+(-1)^(n-j))/2)*Sum_{i=0..j} C(j,i)*C(i,j-k-i).

A322046 Irregular triangle read by rows: a Pascal "rhombus", third in the sequence after A059317 and A027907.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 8, 10, 8, 3, 1, 1, 4, 13, 24, 31, 24, 13, 4, 1, 1, 5, 19, 45, 78, 93, 78, 45, 19, 5, 1, 1, 6, 26, 74, 158, 248, 290, 248, 158, 74, 26, 6, 1

Offset: 0

N. J. A. Sloane, Dec 07 2018

			Triangle begins:
  1,
  1,1,1,
  1,2,4,2,1,
  1,3,8,10,8,3,1,,
  1,4,13,24,31,24,13,4,1,,
  1,5,19,45,78,93,78,45,19,5,1,,
  1,6,26,74,158,248,290,248,158,74,26,6,1,
  ...

Sheng-Liang Yang and Yuan-Yuan Gao, The Pascal rhombus and Riordan arrays, Fib. Q., 56:4 (2018), 337-347. See Fig. 4.

Cf. A059317 and A027907.

cp's OEIS Frontend

A059345 Central column of Pascal's "rhombus" (actually a triangle) A059317.

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A106053 Next-to-central column of triangle in A059317.

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A106050 Column two-from-center of triangle A059317.

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A106058 4th diagonal of triangle in A059317.

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A267192 Column 3 of triangle in A059317 (the Pascal "Rhombus").

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A106113 5th diagonal of triangle in A059317.

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A106150 6th diagonal of triangle in A059317.

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A106173 7th diagonal of triangle in A059317.

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