A059345 -id:A059345 - cp's OEIS frontend

A059317 Pascal's "rhombus" (actually a triangle T(n,k), n >= 0, 0<=k<=2n) read by rows: each entry is sum of 3 terms above it in previous row and one term above it two rows back.

1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 8, 9, 8, 3, 1, 1, 4, 13, 22, 29, 22, 13, 4, 1, 1, 5, 19, 42, 72, 82, 72, 42, 19, 5, 1, 1, 6, 26, 70, 146, 218, 255, 218, 146, 70, 26, 6, 1, 1, 7, 34, 107, 261, 476, 691, 773, 691, 476, 261, 107, 34, 7, 1, 1, 8, 43, 154, 428, 914, 1574, 2158

Offset: 0

N. J. A. Sloane, Jan 26 2001

The rows have lengths 1, 3, 5, 7, ...; cf. A005408.
T(n,k) is the number of paths in the right half-plane from (0,0) to (n,k-n), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: T(3,4)=8 because we have hhU, HU, hUh, Uhh, UH, DUU, UDU and UUD. Row sums yield A006190. - Emeric Deutsch, Sep 03 2007
Let p(n,x) denote the Fibonacci polynomial, defined by p(1,x) = 1, p(2,x) = x, p(n,x) = x*p(n-1,x) + p(n-2,x). The coefficients of the numerator polynomial of the rational function p(n, x + 1 + 1/x) form row n of the triangle A059317; the first three numerator polynomials are 1, 1 + x + x^2, 1 + 2*x + 4*x^2 + 2*x^3 + x^4. - Clark Kimberling, Nov 04 2013

			Triangle begins:
  1;
  1, 1, 1;
  1, 2, 4, 2, 1;
  1, 3, 8, 9, 8, 3, 1;
  ...

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

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Steven R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.
J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 231-236.
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121.
José L. Ramírez, The Pascal Rhombus and the Generalized Grand Motzkin Paths, arXiv:1511.04577 [math.CO], 2015.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv:1504.04404 [math.CO], 2015.
Sheng-Liang Yang and Yuan-Yuan Gao, The Pascal rhombus and Riordan arrays, Fib. Q., 56:4 (2018), 337-347.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A059318, A007318. Row sums give A006190. Central column is A059345.
Other columns: A106050, A106053, A034856, A106058, A106113, A106150, A106173, A267192.
Cf. also A006190, A140750.

-- import Data.List (zipWith4)
a059317 n k = a059317_tabf !! n !! k
a059317_row n = a059317_tabf !! n
a059317_tabf = [1] : [1,1,1] : f [1] [1,1,1] where
   f ws vs = vs' : f vs vs' where
     vs' = zipWith4 (\r s t x -> r + s + t + x)
           (vs ++ [0,0]) ([0] ++ vs ++ [0]) ([0,0] ++ vs)
           ([0,0] ++ ws ++ [0,0])
-- Reinhard Zumkeller, Jun 30 2012

r:=proc(i,j) option remember; 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(seq(r(i,j),j=-i+1..i-1),i=0..9); # Emeric Deutsch, Jun 06 2004
g:=1/(1-z-z*w-z*w^2-z^2*w^2): gser:=simplify(series(g,z=0,10)): for n from 0 to 8 do P[n]:=sort(coeff(gser,z,n)) end do: for n from 0 to 8 do seq(coeff(P[n],w,k),k=0..2*n) end do; # yields sequence in triangular form; 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; Flatten[ Table[ t[n, k], {n, 0, 8}, {k, 0, 2n}]] (* Jean-François Alcover, Feb 01 2012 *)

T(n+1, k) = T(n, k-1) + T(n, k) + T(n, k+1) + T(n-1, k).
Another definition: T(i, j) is defined for i >= 0, -infinity <= j <= infinity; T(i, j) = T(i-1, j) + T(i-1, j-1) + T(i-1, j-2) + T(i-2, j-2) for i >= 2, all j; T(0, 0) = T(1, 1) = T(1, 1) = T(1, 2) = 1; T(0, j) = 0 for j != 0; T(1, j) = 0 for j != 0, 1, 2.
G.f.: Sum_{n>=0, k=0..2*n} T(n, k)*z^n*w^k = 1/(1-z-z*w-z*w^2-z^2*w^2).
There does not seem to be a simple expression for T(n, k). [That may have been true in 2001, but it is no longer true, as the following formulas show. - N. J. A. Sloane, Jan 22 2016]
If the rows of the sequence are displayed in the shape of an isosceles triangle, then, for k>=0, columns k and -k have g.f. z^k*g^k/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
T(i,j) = Sum_{m=0..i} Sum_{l=0..i-j-2*m} binomial(2*m+j,m)*binomial(l+j+2*m,l)*binomial(l,i-j-2*m-l) (see Ramirez link). - José Luis Ramírez Ramírez, Nov 18 2015
The e.g.f of the j-th column of the Pascal rhombus is L_j(x)=(F(x)^(j+1)*C(F(x)^2)^j)/(x*(1-2*F(x)^2*C(F(x)^2))), where F(x) and C(x) are the generating function of the Fibonacci numbers and Catalan numbers. - José Luis Ramírez Ramírez, Nov 18 2015

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

A132885 Triangle read by rows: T(n,k) is the 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), having k H=(2,0) steps (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 3, 1, 7, 2, 19, 9, 1, 51, 28, 3, 141, 95, 18, 1, 393, 306, 70, 4, 1107, 987, 285, 30, 1, 3139, 3144, 1071, 140, 5, 8953, 9963, 3948, 665, 45, 1, 25653, 31390, 14148, 2856, 245, 6, 73789, 98483, 49815, 11844, 1330, 63, 1, 212941, 307836, 172645, 47160

Offset: 0

Emeric Deutsch, Sep 03 2007

Row n has 1+floor(n/2) terms. T(n,0)=A002426(n) (the central trinomial coefficients). T(n,1)=A109188(n-1). Row sums yield A059345. See A132280 for the same statistic on paths restricted to the first quadrant.

			T(4,1)=9 because we have hhH, hHh, Hhh, HUD, UDH, UHD, HDU, DUH and DHU.
Triangle starts:
                     1;
                     1;
                 3,      1;
                 7,      2;
            19,      9,      1;
            51,     28,      3;
       141,     95,     18,      1;
       393,    306,     70,      4;
  1107,    987,    285,     30,      1;
  3139,   3144,   1071,    140,      5;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Columns k=0..3 give A002426, A109188(n-1), A373651(n-4), A375260(n-6).
Row sums gives A059345.
Cf. A132280.

G:=1/sqrt((1+z-t*z^2)*(1-3*z-t*z^2)): Gser:=simplify(series(G,z=0,18)): for n from 0 to 13 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
A132885 := (n,k) -> binomial(n-k,k)*hypergeom([k-n/2,k-n/2+1/2], [1], 4): seq(print(seq(round(evalf(A132885(n,k))),k=0..iquo(n,2))),n=0..9); # Peter Luschny, Sep 18 2014

T[n_, k_] := Binomial[n - k, k]*Hypergeometric2F1[k - n/2, k - n/2 + 1/2, 1, 4]; Table[T[n, k], {n,0,10}, {k, 0, Floor[n/2]}] // Flatten  (* G. C. Greubel, Mar 01 2017 *)

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

T(n,k) = C(n-k,k)*hypergeom([k-n/2,k-n/2+1/2], [1], 4). - Peter Luschny, Sep 18 2014

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

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).

A132884 Triangle read by rows: T(n,k) is the 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), having k h=(1,0) steps (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 0, 8, 0, 1, 13, 0, 15, 0, 1, 0, 57, 0, 24, 0, 1, 63, 0, 156, 0, 35, 0, 1, 0, 384, 0, 340, 0, 48, 0, 1, 321, 0, 1380, 0, 645, 0, 63, 0, 1, 0, 2505, 0, 3800, 0, 1113, 0, 80, 0, 1, 1683, 0, 11145, 0, 8855, 0, 1792, 0, 99, 0, 1, 0, 16008, 0, 37065, 0, 18368, 0, 2736, 0

Offset: 0

Emeric Deutsch, Sep 03 2007

T(2n,0)=A001850(n) (the central Delannoy numbers); T(2n+1,0)=0. T(2n,1)=0; T(2n-1,1)=A108666(n). T(n,k)=0 if n+k is odd. Row sums yield A059345. See A132277 for the same statistic on paths restricted to the first quadrant.

			Triangle starts:
   1;
   0,  1;
   3,  0,  1;
   0,  8,  0,  1;
  13,  0, 15,  0,  1;
   0, 57,  0, 24,  0,  1;
T(3,1)=8 because we have hH, Hh, hUD, UhD, UDh, hDU, DhU and DUh.

Cf. A001850, A108666, A059345, A132277.

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

G.f. = 1/sqrt((1-tz-z^2)^2-4z^2).

A132886 Triangle read by rows: T(n,k) is the 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), having k U steps (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 5, 18, 6, 8, 44, 30, 13, 102, 120, 20, 21, 222, 390, 140, 34, 466, 1140, 700, 70, 55, 948, 3066, 2800, 630, 89, 1884, 7770, 9800, 3780, 252, 144, 3672, 18780, 31080, 17850, 2772, 233, 7044, 43710, 91560, 72450, 19404, 924, 377, 13332, 98610

Offset: 0

Emeric Deutsch, Sep 03 2007

Row n has 1+floor(n/2) terms. T(n,0) = A000045(n+1) (the Fibonacci numbers). T(2n,n) = binomial(2n,n) = A000984(n) (the central binomial coefficients). Row sums yield A059345. Column k has g.f. = binomial(2k,k)*z^(2k)/(1-z-z^2)^(2k+1); accordingly, T(n,1) = 2*A001628(n-2), T(n,2) = 6*A001873(n-4), T(n,3) = 20*A001875(n-6). See A132883 for the same statistic on paths restricted to the first quadrant.

			Triangle starts:
   1;
   1;
   2,   2;
   3,   6;
   5,  18,   6;
   8,  44,  30;
  13, 102, 120,  20;
T(3,1)=6 because we have hUD, UhD, UDh, hDU, DhU and DUh.

Cf. A000045, A059345, A001628, A001873, A001875, A132883.

G:=1/sqrt((1-z-z^2)^2-4*t*z^2): Gser:=simplify(series(G,z=0,17)): for n from 0 to 13 do P[n]:= sort(coeff(Gser,z,n)) end do: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

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

cp's OEIS Frontend

A059317 Pascal's "rhombus" (actually a triangle T(n,k), n >= 0, 0<=k<=2n) read by rows: each entry is sum of 3 terms above it in previous row and one term above it two rows back.

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A132885 Triangle read by rows: T(n,k) is the 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), having k H=(2,0) steps (0 <= k <= floor(n/2)).

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A106050 Column two-from-center of triangle A059317.

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A160905 Right hand side of Pascal rhombus A059317.

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A132884 Triangle read by rows: T(n,k) is the 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), having k h=(1,0) steps (0<=k<=n).

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A132886 Triangle read by rows: T(n,k) is the 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), having k U steps (0 <= k <= floor(n/2)).

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