A005773 -id:A005773 - cp's OEIS frontend

A344507 a(n) = [x^n] 2/(3x + sqrt((1 - 3x)*(x + 1)) + 1).

1, -1, 2, -2, 5, -3, 15, 3, 59, 73, 308, 632, 1951, 4829, 13674, 36306, 100827, 275493, 765150, 2120466, 5918943, 16547595, 46452387, 130703031, 368825661, 1043125407, 2957013140, 8399389528, 23904802109, 68154435941, 194639738503, 556733127851, 1594781146419

Offset: 0

Peter Luschny, May 23 2021

sign

Taras Goy and Mark Shattuck, Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.

Cf. A005043, A001006, A005773, A059738, A344506, A330799.

gf := 2/(3*x + sqrt((1 - 3*x)*(x + 1)) + 1):
ser := series(gf, x, 27): seq(coeff(ser, x, n), n = 0..25);
# Or:
rgf := (x - 2*x^2) / (3*x^2 - 3*x + 1):
subsop(1 = NULL, gfun:-seriestolist(series(rgf, x, 32), 'revogf'));

a[n_] := Sum[(-2)^k Binomial[n, k] Hypergeometric2F1[(k - n)/2, (k - n + 1)/2, k + 2, 4], {k, 0, n}]; Table[a[n], {n, 0, 32}]
(* Or: *)
rgf := (x - 2 x^2) / (3 x^2 - 3 x + 1);
CoefficientList[InverseSeries[Series[rgf, {x, 0, 32}]] / x, x]

R. = PowerSeriesRing(QQ, default_prec=32)
f = (x - 2*x^2) / (3*x^2 - 3*x + 1)
f.reverse().shift(-1).list()

a(n) = [x^n] reverse((x - 2*x^2) / (3*x^2 - 3*x + 1)) / x.
a(n) = Sum_{k=0..n}(-2)^k*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).
a(n) = (9*(n - 2)*a(n - 3) + (12*n - 15)*a(n - 2) + (n - 5)*a(n - 1))/(2*n + 2) for n >= 3.

A059715 Number of multi-directed animals on the triangular lattice.

Original entry on oeis.org

1, 3, 11, 44, 184, 790, 3450, 15242, 67895, 304267, 1369761, 6188002, 28031111, 127253141, 578694237, 2635356807, 12015117401, 54831125131, 250418753498, 1144434017309

Offset: 1

Mireille Bousquet-Mélou, Feb 08 2001

Counts certain animals that generalize directed animals. They are also equinumerous with a class of n-ominoes studied by Klarner in 1967.

M. Bousquet-Mélou and A. Rechnitzer, Lattice animals and heaps of dimers
M. Bousquet-Mélou and A. Rechnitzer, Lattice animals and heaps of dimers, Discrete Math. 258 (2002), no. 1-3, 235-274.
J.-P. Bultel and S. Giraudo, Combinatorial Hopf algebras from PROs, arXiv preprint arXiv:1406.6903 [math.CO], 2014.
D. A. Klarner, Cell growth problems, Canad. J. Math. 19 (1967) 851-863.

Cf. A000108, A005773.

terms = 12;
c[g_, t_] := c[g, t] = Sum[c[g, n, t], {n, 0, 2 terms}];
c[g_, n_, t_] := c[g, n, t] = P[g, n, t] - Sum[c[g, k, t] P[g, n-k-1, t], {k, 0, n-1}];
P[g_, n_, t_] := 1/F[g, n, t];
F[g_, n_, t_] := F[g, n, t] = If[n<=g, 1, F[g, n-1, t] - t F[g, n-g-1, t]];
Rest[CoefficientList[1-1/c[1, t] + O[t]^(terms+1), t]][[1 ;; terms]] (* Jean-François Alcover, Jul 25 2018 *)

The generating function is known in closed form. It is big and non-D-finite.
Bultel-Giraudo (2014), Prop. 3.2, give a g.f. - N. J. A. Sloane, Sep 21 2014
Conjecture: a(n) = Sum_{j=0..n-1} R(n-1, j) for n > 0 where R(n, j) = Sum_{p=0..n - j - 1} binomial(j + p + 2, p + 1)*R(n - j - 1, p) for 0 <= j < n with R(n, n) = 1. - Mikhail Kurkov, Aug 09 2023
a(n) ~ c * d^n, where d = 4.5878943629412631496341355193804435266001072071... and  c = 0.0653089423402623226212483954648487116904937... - Vaclav Kotesovec, Aug 13 2023

A135308 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k DUUU's.

Original entry on oeis.org

1, 1, 2, 5, 13, 1, 35, 7, 96, 36, 267, 159, 3, 750, 645, 35, 2123, 2475, 264, 6046, 9136, 1602, 12, 17303, 32773, 8515, 195, 49721, 115017, 41349, 1925, 143365, 396730, 188010, 14740, 55, 414584, 1349440, 813072, 96200, 1144, 1201917, 4537368

Offset: 0

N. J. A. Sloane, Dec 07 2007

Row n has floor((n+2)/3) terms (n>=1). Row sums yield the Catalan numbers (A000108). Column 0 yields A005773. - Emeric Deutsch, Dec 13 2007

			Triangle begins:
  1
  1
  2
  5
  13 1
  35 7
  96 36
  267 159 3
  ...
T(5,1) = 7 because we have UDUUUUDDDD, UDUUUDUDD, UDUUUDDUDD, UDUUUDDDUD, UDUDUUUDDD, UUDUUUDDDD and UUDDUUUDDD.

Alois P. Heinz, Rows n = 0..250, flattened
FindStat - Combinatorial Statistic Finder, The number of occurrences of the contiguous pattern [.,[[[.,.],.],.]].
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Cf. A000108, A005773.

T:=proc(n,k) options operator, arrow: binomial(n,k)*(sum((-1)^(j-k+1)*3^(n-j)*binomial(n-k,j-k)*binomial(2*j-2-3*k,j-1),j=3*k+1..n))/n end proc: 1; for n to 15 do seq(T(n,k),k=0..floor((n-1)*1/3)) end do; # yields sequence in triangular form; Emeric Deutsch, Dec 13 2007
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, expand(b(x-1, y+1, [1, 3, 4, 1][t])
      * `if`(t=4, z, 1) +b(x-1, y-1, [2, 2, 2, 2][t]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jun 10 2014

T[n_, k_] := (1/n)*Binomial[n, k]*Sum[(-1)^(j-k+1)*3^(n-j)*Binomial[n-k, j-k]*Binomial[2j-2-3k, j-1], {j, 3k+1, n}]; T[0, 0] = 1; Table[T[n, k], {n, 0, 15}, {k, 0, If[n == 0, 0, Quotient[n-1, 3]]}] // Flatten (* Jean-François Alcover, Nov 27 2014, after Emeric Deutsch *)

From Emeric Deutsch, Dec 13 2007: (Start)
T(n,k) = (1/n)*C(n,k)*Sum_{j=3*k+1..n} (-1)^(j-k+1)*3^(n-j)*C(n-k,j-k)*C(2*j-2-3*k,j-1) (n>=1).
G.f.: F=F(t,z) satisfies tzF^3 + [3(1-t)z-1]F^2 - [3(1-t)z-1]F + (1-t)z = 0. (End)

Edited and extended by Emeric Deutsch, Dec 13 2007

A155083 A directed animal related triangle.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 9, 6, 1, 13, 29, 25, 10, 1, 35, 92, 100, 55, 15, 1, 96, 291, 377, 266, 105, 21, 1, 267, 915, 1375, 1169, 602, 182, 28, 1

Offset: 0

Paul Barry, Jan 19 2009

First column is A005773. Row sums are A129775.

			Triangle begins
1,
1, 1,
2, 3, 1,
5, 9, 6, 1,
13, 29, 25, 10, 1,
35, 92, 100, 55, 15, 1,
96, 291, 377, 266, 105, 21, 1,
267, 915, 1375, 1169, 602, 182, 28, 1

G.f.: 1/(1-xy-x/(1-x-xy-x^2/(1-x-xy-x^2/(1-x-xy-x^2/(1-...))))) (continued fraction).

A171505 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A059738.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 34, 29, 9, 1, 117, 128, 57, 12, 1, 405, 538, 309, 94, 15, 1, 1407, 2192, 1533, 604, 140, 18, 1, 4899, 8740, 7179, 3453, 1040, 195, 21, 1, 17083, 34296, 32278, 18264, 6730, 1644, 259, 24, 1, 59629, 132929, 140790, 91372, 39668, 11877, 2443

Offset: 0

Philippe Deléham, Dec 10 2009

Equal to B*A096164 = A171488*B, B=A007318.

			Triangle begins :
1 ;
3, 1 ;
10, 6, 1 ;
34, 29, 9, 1 ;
117, 128, 57, 12, 1 ; ...

Cf. A097609, A064189, A171488, A096164

Sum_{k, 0<=k<=n} T(n,k)*x^k = A005043(n), A001006(n), A005773(n+1), A059738(n) for x = -3, -2, -1, 0 respectively.

T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + sum_{i, i>=0} T(n-1,k+1+i)*(-2)^i. - Philippe Deléham, Feb 23 2012

A178113 Transform of C(n+1,floor((n+1)/2)) by A178112.

Original entry on oeis.org

1, 2, 2, 4, 5, 10, 13, 26, 35, 70, 96, 192, 267, 534, 750, 1500, 2123, 4246, 6046, 12092, 17303, 34606, 49721, 99442, 143365, 286730, 414584, 829168, 1201917, 2403834, 3492117, 6984234, 10165779, 20331558, 29643870, 59287740, 86574831, 173149662

Offset: 0

Paul Barry, May 20 2010

Hankel transform is A178115.

Cf. A178112, A178115, A005773.

a(n) = Sum_{k=0..n} C(floor(n/2),floor(k/2))*((1+(-1)^(n-k))/2)*C(k+1,floor((k+1)/2)). [The formula seems to generate A026392, not these terms. - R. J. Mathar, Feb 10 2015]

Conjecture: a(2*n) = A005773(n+1), a(2*n+1) = 2*a(2*n). - Jason Yuen, Feb 09 2025

A263841 Expansion of (1 - 2x - x^2)/(sqrt(1+x)(1-3x)^(3/2)2x) - 1/(2x).

Original entry on oeis.org

1, 3, 9, 28, 87, 271, 843, 2619, 8123, 25153, 77763, 240054, 740017, 2278329, 7006093, 21520872, 66039651, 202462113, 620164491, 1898109900, 5805127269, 17741909157, 54188530641, 165405964227, 504601360389, 1538559689751, 4688812503053, 14282580916834, 43486805133903

Offset: 0

N. J. A. Sloane, Nov 02 2015

nonn

A. Asinowski and G. Rote, Point sets with many non-crossing matchings, arXiv preprint arXiv:1502.04925 [cs.CG], 2015. See Table 1.

Cf. A005773, A132894, A189911.

A263841 := n -> add((k+1)*binomial(n, k)*binomial(n-k, iquo(n-k,2)), k = 0 .. n):
seq(A263841(n), n = 0 .. 28); # Mélika Tebni, Jan 25 2024

CoefficientList[Series[(1-2x-x^2)/(Sqrt[1+x] (1-3x)^(3/2) 2x)-1/(2x),{x,0,30}],x] (* Harvey P. Dale, Aug 21 2017 *)

my(x='x+O('x^40)); Vec((1-2*x-x^2)/(sqrt(1+x)*(1-3*x)^(3/2)*2*x)-1/(2*x)) \\ Altug Alkan, Nov 10 2015

D-finite with recurrence: -(n+1)*(n^2+n-3)*a(n) + 2*(n^3+3*n^2-4*n-3)*a(n-1) + 3*(n-1)*(n^2+3*n-1)*a(n-2) = 0. - R. J. Mathar, Feb 17 2016
From Mélika Tebni, Jan 24 2024: (Start)
a(n) = A005773(n+1) + A132894(n).
E.g.f.: (1+x)*exp(x)*(BesselI(0,2*x) + BesselI(1,2*x)). (End)
From Mélika Tebni, Jan 25 2024: (Start)
a(n) = Sum_{k=0..n} A189911(k)*binomial(n,k).
a(n) = Sum_{k=0..n} (k+1)*binomial(n,k)*binomial(n-k,floor((n-k)/2)). (End)

A290265 The number of non-palindromic Motzkin paths of length n.

Original entry on oeis.org

0, 0, 0, 2, 4, 16, 38, 114, 288, 800, 2092, 5702, 15244, 41568, 112884, 309822, 851344, 2354656, 6530336, 18193238, 50834716, 142530256, 400713502, 1129710694, 3192584432, 9043259136, 25669403892, 73007358218, 208022076292, 593741582912, 1697381979094, 4859758184274, 13933559180928

Offset: 0

R. J. Mathar, Jul 25 2017

nonn

The Motzkin paths (A001006) are classified here as either palindromic or non-palindromic. The latter are counted by the sequence 1, 1, 2, 2, 3, 3, 5, 5..., offset 0, i.e., entries of A005773 repeated.
Non-palindromic means, there is at least one step of the n (say, the s-th) which does not match the (n-s+1)st step. Not matching means, if the s-th step is U, the (n-s+1)st step is not D, or if the s-th step is F (sometimes also denoted H), the (n-s+1)st step is not F.
All terms are even (because a non-palindromic path reversed creates a different non-palindromic path).

Cf. A001006, A005773.

A290265 := proc(n)
    A001006(n)-A005773(1+floor(n/2)) ;
end proc:
seq( A290265(n),n=0..60) ;

a001006[n_]:=Hypergeometric2F1[(1-n)/2, -n/2, 2, 4]; a005773[n_]:=If[n==0, 1, Sum[k*Sum[Binomial[n, j]*Binomial[j, 2*j-n-k], {j, 0, n}]/n, {k, 1, n}]]; Table[a001006[n] - a005773[1 + Floor[n/2]], {n, 0, 50}] (* Indranil Ghosh, Aug 04 2017 *)

a(n) = A001006(n)-A005773(1+floor(n/2)).

Conjecture: -(16*n-47)*(n+2)*(n+1)*a(n) -(n+1)*(9*n^2-167*n+188)*a(n-1) +n*(139*n^2-450*n+59)*a(n-2) +(n-1)*(187*n^2-1619*n+2250)*a(n-3) -(n-2)*(97*n^2+346*n-2255)*a(n-4) +(-311*n^3+2398*n^2-5779*n+4112)*a(n-5) +3*(-153*n^3+1458*n^2-4361*n+4348)*a(n-6) -3*(n-6)*(169*n^2-1152*n+1799)*a(n-7) -9*(n-6)*(n-7)*(23*n-82)*a(n-8)=0.

A292630 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)(BesselI(0,2x) + BesselI(1,2x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 10, 13, 6, 1, 5, 17, 35, 35, 10, 1, 6, 26, 75, 126, 96, 20, 1, 7, 37, 139, 339, 462, 267, 35, 1, 8, 50, 233, 758, 1558, 1716, 750, 70, 1, 9, 65, 363, 1491, 4194, 7247, 6435, 2123, 126, 1, 10, 82, 535, 2670, 9660, 23460, 34016, 24310, 6046, 252, 1, 11, 101, 755, 4451, 19846, 63195, 132339, 160795, 92378, 17303, 462

Offset: 0

Ilya Gutkovskiy, Sep 20 2017

A(n,k) is the k-th binomial transform of A001405 evaluated at n.

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 2)*x^2/2! +  (k^3 + 3*k^2 + 6*k + 3)*x^3/3! + (k^4 + 4*k^3 + 12*k^2 + 12*k + 6)*x^4/4! + ...
Square array begins:
   1,   1,    1,     1,     1,     1,  ...
   1,   2,    3,     4,     5,     6,  ...
   2,   5,   10,    17,    26,    37,  ...
   3,  13,   35,    75,   139,   233,  ...
   6,  35,  126,   339,   758,  1491,  ...
  10,  96,  462,  1558,  4194,  9660,  ...

N. J. A. Sloane, Transforms

Columns k=0..5 give A001405, A005773 (with first term deleted), A001700, A026378 (with offset 0), A005573, A122898.

Main diagonal gives A292631.

[seq(seq((k)!*add((m-j)^(j-i)/floor(i/2)!/ceil(i/2)!/(j-i)!,i=0..j),j=0..m), m=0..20)]; # Robert Israel, Sep 20 2017

Table[Function[k, n! SeriesCoefficient[Exp[k x] (BesselI[0, 2 x] + BesselI[1, 2 x]), {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(k*x)*(BesselI(0,2*x) + BesselI(1,2*x)).

A296449 Triangle I(m,n) read by rows: number of perfect lattice paths on the m*n board.

Original entry on oeis.org

1, 2, 4, 3, 7, 17, 4, 10, 26, 68, 5, 13, 35, 95, 259, 6, 16, 44, 122, 340, 950, 7, 19, 53, 149, 421, 1193, 3387, 8, 22, 62, 176, 502, 1436, 4116, 11814, 9, 25, 71, 203, 583, 1679, 4845, 14001, 40503, 10, 28, 80, 230, 664, 1922, 5574, 16188, 47064, 136946, 11, 31, 89, 257, 745, 2165, 6303, 18375, 53625, 156629, 457795

Offset: 1

R. J. Mathar, Dec 13 2017

			Triangle begins:
   1;
   2,  4;
   3,  7, 17;
   4, 10, 26,  68;
   5, 13, 35,  95, 259;
   6, 16, 44, 122, 340,  950;
   7, 19, 53, 149, 421, 1193, 3387;
   8, 22, 62, 176, 502, 1436, 4116, 11814;
   9, 25, 71, 203, 583, 1679, 4845, 14001, 40503;
  10, 28, 80, 230, 664, 1922, 5574, 16188, 47064, 136946;

D. Yaqubi, M. Farrokhi D. G., and H. Ghasemian Zoeram, Lattice paths inside a table, I, arXiv:1612.08697 [math.CO], 2016-2017, array I(m,n).

Cf. A081113 (diagonal), A000079 (2nd row), A001333 (3rd row), A126358, A057960, A126360, A002714, A126362, A188866.

Inm := proc(n,m)
    if m >= n then
        (n+2)*3^(n-2)+(m-n)*add(A005773(i)*A005773(n-i),i=0..n-1)
            +2*add((n-k-2)*3^(n-k-3)*A001006(k),k=0..n-3) ;
    else
        0 ;
    end if;
end proc:
for m from 1 to 13 do
for n from 1 to m do
    printf("%a,",Inm(n,m)) ;
end do:
printf("\n") ;
end do:
# Second program:
A296449row := proc(n) local gf, ser;
gf := n -> 1 + (x*(n - (3*n + 2)*x) + (2*x^2)*(1 +
ChebyshevU(n - 1, (1 - x)/(2*x))) / ChebyshevU(n, (1 - x)/(2*x)))/(1 - 3*x)^2;
ser := n -> series(expand(gf(n)), x, n + 1);
seq(coeff(ser(n), x, k), k = 1..n) end:
for n from 0 to 11 do A296449row(n) od; # Peter Luschny, Sep 07 2021

(* b = A005773 *) b[0] = 1; b[n_] := Sum[k/n*Sum[Binomial[n, j] * Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}];
(* c = A001006 *) c[0] = 1; c[n_] := c[n] = c[n-1] + Sum[c[k] * c[n-2-k], {k, 0, n-2}];
Inm[n_, m_] := If[m >= n, (n + 2)*3^(n - 2) + (m - n)*Sum[b[i]*b[n - i], {i, 0, n - 1}] + 2*Sum[(n - k - 2)*3^(n - k - 3)*c[k], {k, 0, n-3}], 0];
Table[Inm[n, m], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 23 2018, adapted from first Maple program. *)

I(m,n) = (n+2)*3^(n-2) + (m-n)*Sum_{i=0..n-1} A005773(i)*A005773(n-i) + 2*Sum_{k=0..n-3} (n-k-2)*3^(n-k-3)*A001006(k). [Yaqubi Corr. 2.10]

I(m,n) = A188866(m-1,n) for m > 1. - Pontus von Brömssen, Sep 06 2021

cp's OEIS Frontend

A344507 a(n) = [x^n] 2/(3*x + sqrt((1 - 3*x)*(x + 1)) + 1).

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A059715 Number of multi-directed animals on the triangular lattice.

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A135308 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k DUUU's.

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A155083 A directed animal related triangle.

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A171505 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A059738.

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A178113 Transform of C(n+1,floor((n+1)/2)) by A178112.

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A263841 Expansion of (1 - 2*x - x^2)/(sqrt(1+x)*(1-3*x)^(3/2)*2*x) - 1/(2*x).

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A290265 The number of non-palindromic Motzkin paths of length n.

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A292630 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*(BesselI(0,2*x) + BesselI(1,2*x)).

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A344507 a(n) = [x^n] 2/(3x + sqrt((1 - 3x)*(x + 1)) + 1).

A263841 Expansion of (1 - 2x - x^2)/(sqrt(1+x)(1-3x)^(3/2)2x) - 1/(2x).

A292630 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)(BesselI(0,2x) + BesselI(1,2x)).