A137644 -id:A137644 - cp's OEIS frontend

A063967 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 5, 15, 16, 7, 1, 8, 30, 43, 29, 9, 1, 13, 58, 104, 95, 46, 11, 1, 21, 109, 235, 271, 179, 67, 13, 1, 34, 201, 506, 705, 591, 303, 92, 15, 1, 55, 365, 1051, 1717, 1746, 1140, 475, 121, 17, 1, 89, 655, 2123, 3979, 4759, 3780, 2010, 703, 154, 19, 1

Offset: 0

Henry Bottomley, Sep 05 2001

			T(3,1) = T(2,1) + T(1,1) + T(2,0) + T(1,0) = 3 + 1 + 2 + 1 = 7.
Triangle begins:
   1,
   1,   1,
   2,   3,   1,
   3,   7,   5,   1,
   5,  15,  16,   7,   1,
   8,  30,  43,  29,   9,   1,
  13,  58, 104,  95,  46,  11,  1,
  21, 109, 235, 271, 179,  67, 13,  1,
  34, 201, 506, 705, 591, 303, 92, 15, 1

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
Emanuele Munarini, A generalization of André-Jeannin's symmetric identity, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 98-118.

Row sums are A002605.
Columns include: A000045(n+1), A023610(n-1).
Main diagonal: A000012, a(n, n-1) = A005408(n-1).
Matrix inverse: A091698, matrix square: A091700.
Cf. A321620.
Sum_{k=0..n} x^k*T(n,k) is (-1)^n*A057086(n) (x=-11), (-1)^n*A057085(n+1) (x=-10), (-1)^n*A057084(n) (x=-9), (-1)^n*A030240(n) (x=-8), (-1)^n*A030192(n) (x=-7), (-1)^n*A030191(n) (x=-6), (-1)^n*A001787(n+1) (x=-5), A000748(n) (x=-4), A108520(n) (x=-3), A049347(n) (x=-2), A000007(n) (x=-1), A000045(n) (x=0), A002605(n) (x=1), A030195(n+1) (x=2), A057087(n) (x=3), A057088(n) (x=4), A057089(n) (x=5), A057090(n) (x=6), A057091(n) (x=7), A057092(n) (x=8), A057093(n) (x=9). - Philippe Deléham, Nov 03 2006

a063967_tabl = [1] : [1,1] : f [1] [1,1] where
   f us vs = ws : f vs ws where
     ws = zipWith (+) ([0] ++ us ++ [0]) $
          zipWith (+) (us ++ [0,0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Apr 17 2013

T[n_, k_] := Sum[Binomial[j, n - j]*Binomial[j, k], {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 11 2017, after Paul Barry *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[1/(1 - x - x^2), 11] // Flatten (* Peter Luschny, Nov 27 2018 *)

G.f.: 1/(1-x*(1+x)*(1+y)). - Vladeta Jovovic, Oct 11 2003
Riordan array (1/(1-x-x^2), x(1+x)/(1-x-x^2)). The inverse of the signed version (1/(1+x-x^2),x(1-x)/(1+x-x^2)) is abs(A091698). - Paul Barry, Jun 10 2005
T(n, k) = Sum_{j=0..n} C(j, n-j)C(j, k). - Paul Barry, Nov 09 2005
Diagonal sums are A002478. - Paul Barry, Nov 09 2005
A026729*A007318 as infinite lower triangular matrices. - Philippe Deléham, Dec 11 2008
Central coefficients T(2*n,n) are A137644. - Paul Barry, Apr 15 2010
Product of Riordan arrays (1, x(1+x))*(1/(1-x), x/(1-x)), that is, A026729*A007318. - Paul Barry, Mar 14 2011
Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 12 2011

A191354 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), and (2,1).

Original entry on oeis.org

1, 1, 3, 9, 25, 75, 227, 693, 2139, 6645, 20757, 65139, 205189, 648427, 2054775, 6526841, 20775357, 66251247, 211617131, 676930325, 2168252571, 6953348149, 22322825865, 71735559255, 230735316795, 742773456825, 2392949225565, 7714727440755, 24888317247705, 80341227688095

Offset: 0

Joerg Arndt, Jun 30 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1-2*x-3*x^2-4*x^3) )); // G. C. Greubel, Feb 18 2019

a[n_]:= Sum[Binomial[2k, k]*Sum[Binomial[j, n-k-j]*Binomial[k, j]*2^(j-k) *3^(-n+k+2j)*4^(n-k-2j), {j, 0, k}], {k, 0, n}];
Array[a, 30, 0] (* Jean-François Alcover, Jul 21 2018, after Vladimir Kruchinin *)
CoefficientList[Series[1/Sqrt[1-2*x-3*x^2-4*x^3], {x, 0, 30}], x] (* G. C. Greubel, Feb 18 2019 *)

a(n):=sum(binomial(2*k,k) * sum(binomial(j,n-k-j) * 2^(j-k) * binomial(k,j) * 3^(-n+k+2*j) * 4^(n-k-2*j),j,0,k),k,0,n); /* Vladimir Kruchinin, Feb 27 2016 */

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

my(x='x+O('x^30)); Vec(1/sqrt(1-2*x-3*x^2-4*x^3)) \\ G. C. Greubel, Feb 18 2019

(1/sqrt(1-2*x-3*x^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019

G.f.: 1/sqrt(1-2*x-3*x^2-4*x^3). - Mark van Hoeij, Apr 16 2013
G.f.: Q(0), where Q(k) = 1 + x*(2+3*x+4*x^2)*(4*k+1)/( 4*k+2 - x*(2+3*x+4*x^2)*(4*k+2)*(4*k+3)/(x*(2+3*x+4*x^2)*(4*k+3) + 4*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013
a(n) = Sum_{k=0..n} (binomial(2*k,k) * Sum_{j=0..k} (binomial(j,n-k-j) *binomial(k,j)*2^(j-k)*3^(-n+k+2*j)*4^(n-k-2*j))). - Vladimir Kruchinin, Feb 27 2016
D-finite with recurrence: +(n)*a(n) +(-2*n+1)*a(n-1) +3*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Jan 14 2020

A192368 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (2,0), (0,2), (1,1).

Original entry on oeis.org

1, 1, 6, 19, 94, 396, 1870, 8541, 40284, 189274, 899260, 4281168, 20487156, 98299384, 473118174, 2282322211, 11034087438, 53443135944, 259283934816, 1259795078566, 6129223177272, 29856164309124, 145592506783224, 710686739172096, 3472285996766556, 16979257639328076

Offset: 0

Joerg Arndt, Jul 01 2011

nonn

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

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369.

s := RootOf( 16*x*(3*s+1)*s+(s^2-18*s+1)*(s-1), s):
ogf := -16*(3*s+1)*s^(3/2)/(3*s^4+2*s^3-76*s^2+6*s+1):
series(ogf, x=0, 20); # 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,
       b(x-1, y)+b(x-2, y)+b(x, y-2)+b(x-1, y-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, May 16 2017

a[0, 0] = 1; a[n_, k_] /; n >= 0 && k >= 0 := a[n, k] = a[n, k - 1] + a[n, k - 2] + a[n - 1, k - 1] + a[n - 2, k]; a[, ] = 0;
a[n_] := a[n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Oct 14 2019 *)

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

G.f. -16*(3*s+1)*s^(3/2)/(3*s^4+2*s^3-76*s^2+6*s+1) where s satisfies 16*x*(3*s+1)*s+(s^2-18*s+1)*(s-1) = 0. - Mark van Hoeij, Apr 16 2013

A191649 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (2,2).

Original entry on oeis.org

1, 3, 14, 71, 379, 2082, 11651, 66051, 378064, 2180037, 12644861, 73695358, 431209313, 2531556197, 14904832196, 87970766447, 520337606401, 3083584244460, 18304476242735, 108820740004749, 647817646760368, 3861215365595659, 23039691494489015, 137615812845579390

Offset: 0

Joerg Arndt, Jun 30 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 9, 29.

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A191354.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(x^4+2*x^3-x^2-6*x+1) )); // G. C. Greubel, Apr 29 2019

CoefficientList[Series[1/Sqrt[x^4 + 2 x^3 - x^2 - 6 x + 1], {x, 0, 23}], x] (* Michael De Vlieger, Oct 08 2016 *)

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

my(x='x+O('x^30)); Vec(1/sqrt(x^4+2*x^3-x^2-6*x+1)) \\ G. C. Greubel, Apr 29 2019

(1/sqrt(x^4+2*x^3-x^2-6*x+1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 29 2019

G.f.: 1/sqrt(x^4 +2*x^3 -x^2 -6*x +1). - Mark van Hoeij, Apr 17 2013

D-finite with recurrence: n*a(n) +3*(-2*n+1)*a(n-1) +(-n+1)*a(n-2) +(2*n-3)*a(n-3) +(n-2)*a(n-4)=0. - R. J. Mathar, Oct 08 2016

A192371 Number of lattice paths from (0,0) to (n,n) using steps (1,1), (0,2), (2,0), (0,3), (3,0).

Original entry on oeis.org

1, 1, 3, 9, 25, 87, 307, 1113, 4149, 15605, 59201, 225999, 866449, 3333847, 12865335, 49769689, 192945411, 749396493, 2915432049, 11358771965, 44313108627, 173081422997, 676766482917, 2648843996031, 10376891445525, 40685535827325, 159641884780749, 626849029013919, 2463010645910537, 9683604464279235

Offset: 0

Joerg Arndt, Jul 01 2011

nonn

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

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A192368.

s := RootOf( (s^3-s-1)*(s-1)+x*s*(4-3*s), s);
ogf := sqrt(4*s-3*s^2)*(s^3-4*s^2+2*s+2)/((2*s^2-s-2)*(3*s^3-6*s^2+4*s-2)*(1-x)):
series(ogf, x=0, 30);  # Mark van Hoeij, Apr 17 2013
# second Maple program:
b:= proc(p) b(p):= `if`(p=[0$2], 1, `if`(min(p[])<0, 0,
      add(b(p-l), l=[[1, 1], [0, 2], [2, 0], [0, 3], [3, 0]])))
    end:
a:= n-> b([n$2]):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 18 2014

b[p_List] := b[p] = If[p == {0, 0}, 1, If[Min[p] < 0, 0, Sum[b[p - l], {l, {{1, 1}, {0, 2}, {2, 0}, {3, 0}, {0, 3}}}]]]; a[n_] := b[{n, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

/* same as in A092566 but use */
steps=[[1,1], [2,0], [0,2], [3,0], [0,3]];
/* Joerg Arndt, Jun 30 2011 */

G.f.: sqrt(4*s-3*s^2)*(s^3-4*s^2+2*s+2)/((2*s^2-s-2)*(3*s^3-6*s^2+4*s-2)*(1-x)) where the function s satisfies (s^3-s-1)*(s-1)+x*s*(4-3*s) = 0. - Mark van Hoeij, Apr 17 2013

A192417 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (2,2), (3,3).

Original entry on oeis.org

1, 2, 7, 27, 107, 436, 1810, 7609, 32288, 138009, 593311, 2562725, 11112720, 48347332, 210936119, 922550622, 4043488129, 17755735241, 78099099877, 344033901804, 1517535718392, 6701979806379, 29630948706756, 131136723532257, 580901892464599, 2575423975663301

Offset: 0

Joerg Arndt, Jun 30 2011

nonn

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

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A191354.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1) )); // G. C. Greubel, Apr 29 2019

CoefficientList[Series[1/Sqrt[x^6+2x^5+x^4-2x^3-2x^2-4x+1], {x, 0, 25}], x] (* Michael De Vlieger, Oct 08 2016 *)

/* same as in A092566 but use */
steps=[[0,1], [1,0], [2,2], [3,3]];
/* Joerg Arndt, Jun 30 2011 */

my(x='x+O('x^30)); Vec(1/sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1)) \\ G. C. Greubel, Apr 29 2019

(1/sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 29 2019

G.f.: 1/sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1). - Mark van Hoeij, Apr 17 2013

D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +2*(-n+1)*a(n-2) +(-2*n+3)*a(n-3) +(n-2)*a(n-4) +(2*n-5)*a(n-5) +(n-3)*a(n-6)=0. - R. J. Mathar, Oct 08 2016

A192446 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (3,0), (0,1), (0,3).

Original entry on oeis.org

1, 2, 6, 30, 154, 768, 3906, 20232, 105750, 556328, 2943432, 15646932, 83500126, 447057380, 2400249624, 12918250836, 69674241654, 376489511460, 2037768450480, 11045915485740, 59955446568276, 325821729044784, 1772588671356204, 9653187691115640, 52617711157401186, 287051310425050668

Offset: 0

Joerg Arndt, Jul 01 2011

nonn

Diagonal of rational function 1/(1 - (x + y + x^3 + y^3)). - Gheorghe Coserea, Aug 06 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1344 (first 304 terms from Gheorghe Coserea)

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A192368.

REL := 3*s^2*x^2-(1-2*s^2+2*s^3)*x-s*(s^3+2*s^2+s-1);
ogf := sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3);
series(eval(ogf, s=RootOf(REL,s)),x=0,30);  # Mark van Hoeij, Apr 17 2013
# second Maple program:
b:= proc(x, y) option remember; `if`(y=0, 1, add((p->
      `if`(p[1]<0, 0, b(p[1], p[2])))(sort([x, y]-h)),
        h=[[1, 0], [0, 1], [3, 0], [0, 3]]))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 28 2018

a[0, 0] = 1; a[n_, k_] /; n >= 0 && k >= 0 := a[n, k] = a[n, k-1] + a[n, k-3] + a[n-1, k] + a[n-3, k]; a[, ] = 0;
a[n_] := a[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Oct 06 2019 *)

/* same as in A092566 but use */
steps=[[1,0], [3,0], [0,1], [0,3]];
/* Joerg Arndt, Jun 30 2011 */

seq(N) = {
  my(x='x + O('x^N), d=16*x^6 + 16*x^5 + 16*x^4 - 8*x^3 - 4*x^2 + 1,
     s=serreverse((1 - 2*x^2 + 2*x^3 - sqrt(d))/(6*x^2)));
  Vec(sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3));
};
seq(26) \\ Gheorghe Coserea, Aug 06 2018

G.f.: sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3)  where s is a function satisfying 3*s^2*x^2-(1-2*s^2+2*s^3)*x-s*(s^3+2*s^2+s-1)=0. - Mark van Hoeij, Apr 17 2013
From Gheorghe Coserea, Aug 06 2018: (Start)
G.f. y=A(x) satisfies:
0 = (4*x^3 + 8*x^2 + 4*x - 1)^4*(108*x^3 - 108*x^2 + 36*x - 31)^2*y^8 + 4*(4*x^3 + 8*x^2 + 4*x - 1)^3*(36*x^3 + 36*x^2 - 4*x - 13)*(108*x^3 - 108*x^2 + 36*x - 31)*y^6 + 2*(4*x^3 + 8*x^2 + 4*x - 1)^2*(2160*x^6 + 4320*x^5 + 1872*x^4 - 1784*x^3 - 1576*x^2 + 472*x + 431)*y^4 + 4*(4*x^3 + 8*x^2 + 4*x - 1)*(112*x^6 + 448*x^5 + 688*x^4 + 456*x^3 + 96*x^2 + 40*x + 55)*y^2 + (4*x^3 + 12*x^2 + 12*x + 3)^2.
0 = (4*x^3 + 8*x^2 + 4*x - 1)*(108*x^3 - 108*x^2 + 36*x - 31)*(270*x^4 + 180*x^3 + 144*x^2 - 225*x - 59)*y''' + (1283040*x^9 + 1924560*x^8 + 1080864*x^7 - 1425816*x^6 - 2135376*x^5 + 33048*x^4 + 702468*x^3 + 134520*x^2 + 43892*x + 30575)*y'' + 30*(111780*x^8 + 149040*x^7 + 120960*x^6 - 122094*x^5 - 172206*x^4 - 6012*x^3 + 36615*x^2 + 10298*x - 1541)*y' + 60*(29160*x^7 + 34020*x^6 + 36288*x^5 - 43092*x^4 - 45882*x^3 - 6462*x^2 + 1890*x + 913)*y.
(End)

A191678 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (0,2), (2,2).

Original entry on oeis.org

1, 1, 5, 15, 62, 233, 937, 3729, 15121, 61492, 251942, 1036215, 4279754, 17731181, 73670725, 306823695, 1280574706, 5354602495, 22426876445, 94070238840, 395106054632, 1661489413472, 6994494531010, 29474635716345, 124319047552309, 524797934104312, 2217091297558466, 9373180869094923

Offset: 0

Joerg Arndt, Jun 30 2011

nonn

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A191354.

P := (4*x^6+12*x^5-20*x^3+27*x^2+12*x-4)*A^3-(3*x^2+3*x-3)*A+1;
Q := eval(P, A=A+1):
series(RootOf(Q,A)+1, x=0, 30);  # Mark van Hoeij, Apr 17 2013

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

G.f.: A(x) where (4*x^6+12*x^5-20*x^3+27*x^2+12*x-4)*A(x)^3-(3*x^2+3*x-3)*A(x)+1 = 0. - Mark van Hoeij, Apr 17 2013

cp's OEIS Frontend

A063967 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.

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A191354 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), and (2,1).

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A192368 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (2,0), (0,2), (1,1).

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A191649 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (2,2).

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A192371 Number of lattice paths from (0,0) to (n,n) using steps (1,1), (0,2), (2,0), (0,3), (3,0).

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A192417 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (2,2), (3,3).

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Magma

Mathematica

PARI

PARI

Sage

Formula

A192446 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (3,0), (0,1), (0,3).

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