A001006 -id:A001006 - cp's OEIS frontend

A059347 Difference array of Motzkin numbers A001006 read by antidiagonals.

1, 0, 1, 1, 1, 2, 0, 1, 2, 4, 2, 2, 3, 5, 9, 0, 2, 4, 7, 12, 21, 5, 5, 7, 11, 18, 30, 51, 0, 5, 10, 17, 28, 46, 76, 127, 14, 14, 19, 29, 46, 74, 120, 196, 323, 0, 14, 28, 47, 76, 122, 196, 316, 512, 835, 42, 42, 56, 84, 131, 207, 329, 525, 841, 1353, 2188, 0, 42, 84, 140, 224

Offset: 0

N. J. A. Sloane, Jan 27 2001

Row sums of odd rows (e.g., 4 = 1+1+2 for 3rd row) equal the Motzkin number of next row. Row sums of even rows equal the Motzkin number of the next row - n!/((n/2)!((n/2)+1)!) (i.e., A001006(n) - A000108(n/2) where A000108 are the Catalan numbers). - Gerald McGarvey, Dec 05 2004

			Triangle begins:
1;
0,1;
1,1,2;
0,1,2,4;
2,2,3,5,9;
0,2,4,7,12,21;
5,5,7,11,18,30,51;
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

Top row is A001006, leading diagonals give A000108 (interspersed with 0's), A000108 doubled up, A059348.

max = 12; A001006 = CoefficientList[ Series[ (1-x-(1-2x-3x^2)^(1/2))/(2x^2), {x, 0, max}], x] ; row[0] = A001006; row[n_] := Differences[A001006, n]; Flatten[ Table[ row[n-k][[k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Nov 12 2012, from formula *)

More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001

A116388 Expansion of 1/((1+x(1-M(x)))sqrt(1-2x-3x^2)), M(x) the g.f. of A001006.

Original entry on oeis.org

1, 1, 4, 10, 29, 82, 236, 681, 1975, 5745, 16757, 48982, 143442, 420721, 1235663, 3633453, 10695292, 31511524, 92919758, 274203662, 809719718, 2392579638, 7073684393, 20924387460, 61925598216, 183350728661, 543095661673

Offset: 0

Paul Barry, Feb 12 2006

G. C. Greubel, Table of n, a(n) for n = 0..500

List([0..30], n-> Sum([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-k-j) ))); # G. C. Greubel, May 23 2019

[(&+[ (&+[Binomial(n-k, j-k)*Binomial(j, n-k-j): j in [0..n-k]]) : k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 23 2019

Table[Sum[Sum[Binomial[n-k,j-k]*Binomial[j,n-k-j], {j,0,n-k}], {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, May 23 2019 *)

{a(n) = sum(k=0,n\2, sum(j=0,n-k, binomial(n-k, j-k)*binomial(j,n-k-j)))}; \\ G. C. Greubel, May 23 2019

[sum( sum(binomial(n-k, j-k)*binomial(j,n-k-j) for j in (0..n)) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, May 23 2019

G.f.: 2*x/(sqrt(1-2*x-3*x^2)*(sqrt(1-2*x-3*x^2) -1 +2*x +3*x^2)).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(n-k,j-k)*C(j,n-k-j).
Conjecture: n*a(n) + 7*(-n+1)*a(n-1) + 2*(4*n-9)*a(n-2) + (25*n-58)*a(n-3) + (-18*n+65)*a(n-4) + (-52*n+199)*a(n-5) + (-31*n+135)*a(n-6) + 6*(-n+5)*a(n-7) = 0. - R. J. Mathar, Jun 22 2016

A121399 G.f. satisfies: A(x) = G(x)A(x^2G(x)) where G(x) is the g.f. of the Motzkin numbers (A001006): G = (1 + xG + x^2G^2).

Original entry on oeis.org

1, 1, 3, 6, 17, 42, 114, 302, 827, 2263, 6275, 17468, 48967, 137834, 389738, 1105861, 3148240, 8987989, 25726635, 73808069, 212196040, 611219900, 1763659860, 5097131364, 14752847173, 42757853357, 124080269331, 360493591232

Offset: 0

Paul D. Hanna, Jul 27 2006

nonn

Equals column 0 of triangle A121400.

			A(x) = 1 + x + 3*x^2 + 6*x^3 + 17*x^4 + 42*x^5 + 114*x^6 +...
The g.f. of the Motzkin numbers begins:
G(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 +...

Cf. A121400 (triangle), A121398 (main diagonal), A001006 (Motzkin).

{a(n)=local(F=1+x+x^2,G=serreverse(x/(F+x^2*O(x^n)))/x,H=1+x,A); for(i=0,n,H=G*subst(H,x,x^2*G)+x^2*O(x^n)); A=(x*H-y*subst(H,x,x*y))/(x*subst(F,x,y)-y); polcoeff(polcoeff(A,n,x),0,y)}

A162482 Expansion of (1/(1-x)^3)*M(x/(1-x)^3), M(x) the g.f. of Motzkin numbers A001006.

Original entry on oeis.org

1, 4, 14, 53, 218, 945, 4235, 19441, 90947, 432030, 2078416, 10105435, 49578341, 245131321, 1220218293, 6110131376, 30756858405, 155547919269, 789965192900, 4027121386190, 20600180351659, 105707046807196, 543973305719611

Offset: 0

Paul Barry, Jul 04 2009

nonn

Cf. A001006, A162481.

A162482 := proc(n)
    add(binomial(n+2*k+2,n-k)*A001006(k),k=0..n) ;
end proc:
seq(A162482(n),n=0..40) ; # R. J. Mathar, Feb 10 2015

m[n_] := m[n] = If[n == 0, 1, m[n-1] + Sum[m[k]*m[n-2-k], {k, 0, n-2}]];
a[n_] := Sum[Binomial[n+2k+2, n-k]*m[k], {k, 0, n}];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Apr 04 2024 *)

G.f.: 1/((1-x)^3-x-x^2/((1-x)^3-x-x^2/((1-x)^3-x-x^2/((1-x)^3-x-x^2/(1-... (continued fraction);
a(n) = Sum{k=0..n} C(n+2k+2,n-k)*A001006(k).
Conjecture: (n+2)*a(n) +4*(-2*n-1)*a(n-1) +18*(n-1)*a(n-2) +13*(-2*n+5)*a(n-3) +17*(n-4)*a(n-4) +3*(-2*n+11)*a(n-5) +(n-7)*a(n-6)=0. - R. J. Mathar, Feb 10 2015

A200538 Product of Jacobsthal and Motzkin numbers: a(n) = A001045(n+1)*A001006(n).

Original entry on oeis.org

1, 1, 6, 20, 99, 441, 2193, 10795, 55233, 284735, 1494404, 7914270, 42360541, 228460935, 1241224182, 6784445340, 37288826697, 205937705799, 1142317727466, 6361104740100, 35548154733969, 199295884785459, 1120615326442269, 6318077793648075, 35710056983891367, 202297486497822121

Offset: 0

Paul D. Hanna, Nov 18 2011

nonn

The g.f. for the Jacobsthal numbers is 1/(1-x-2*x^2) and the g.f. M(x) for the Motzkin numbers satisfy: M(x) = 1 + x*M(x) + x^2*M(x)^2.

			G.f.: A(x) = 1 + x + 6*x^2 + 20*x^3 + 99*x^4 + 441*x^5 + 2193*x^6 +...
where A(x) = 1*1 + 1*1*x + 3*2*x^2 + 5*4*x^3 + 11*9*x^4 + 21*21*x^5 + 43*51*x^6 + 85*127*x^7 + 171*323*x^8 +...+ A001045(n+1)*A001006(n)*x^n +...

Cf. A200375, A200539, A200540, A001045, A001006.

{A001006(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2),n)}
{A001045(n)=polcoeff( x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=A001045(n+1)*A001006(n)}

A200539 Product of Fibonacci and Motzkin numbers: a(n) = A000045(n+1)*A001006(n).

Original entry on oeis.org

1, 1, 4, 12, 45, 168, 663, 2667, 10982, 45925, 194732, 834912, 3614063, 15771795, 69316740, 306534564, 1362986799, 6089916936, 27328613142, 123118156260, 556626199974, 2524659817449, 11484671681511, 52384730922720, 239534402969925, 1097805759803893, 5042014405418968

Offset: 0

Paul D. Hanna, Nov 18 2011

nonn

The g.f. for the Fibonacci numbers is 1/(1-x-x^2) and the g.f. M(x) for the Motzkin numbers satisfies: M(x) = 1 + x*M(x) + x^2*M(x)^2.

			G.f.: A(x) = 1 + x + 4*x^2 + 12*x^3 + 45*x^4 + 168*x^5 + 663*x^6 +...
where A(x) = 1*1 + 1*1*x + 2*2*x^2 + 3*4*x^3 + 5*9*x^4 + 8*21*x^5 + 13*51*x^6 + 21*127*x^7 + 34*323*x^8 +...+ A000045(n+1)*A001006(n)*x^n +...

Cf. A098614, A200538, A200540, A098616.

{A001006(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2),n)}
{a(n)=fibonacci(n+1)*A001006(n)}

A200540 Product of Pell and Motzkin numbers: a(n) = A000129(n+1)*A001006(n).

Original entry on oeis.org

1, 2, 10, 48, 261, 1470, 8619, 51816, 318155, 1985630, 12561308, 80360280, 519013571, 3379514970, 22161470850, 146227235904, 970126550763, 6467496499590, 43304243215638, 291087137589552, 1963598081845335, 13288619577124122, 90195242361688863, 613843707553183800

Offset: 0

Paul D. Hanna, Nov 18 2011

nonn

The g.f. for the Pell numbers is 1/(1-2*x-x^2) and the g.f. M(x) for the Motzkin numbers satisfy: M(x) = 1 + x*M(x) + x^2*M(x)^2.

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 48*x^3 + 261*x^4 + 1470*x^5 + 8619*x^6 +...
where A(x) = 1*1 + 2*1*x + 5*2*x^2 + 12*4*x^3 + 29*9*x^4 + 70*21*x^5 + 169*51*x^6 + 408*127*x^7 + 985*323*x^8 +...+ A000129(n+1)*A001006(n)*x^n +...

Cf. A098616, A200538, A200539, A098614.

{A001006(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2),n)}
{A000129(n)=polcoeff( x/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=A000129(n+1)*A001006(n)}

A251570 G.f. A(x) satisfies the condition that M(A(x)) is a power series in x consisting entirely of positive integer coefficients such that M(A(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + xM(x) + xM(x)^2 is the g.f. of the Motzkin numbers A001006.

Original entry on oeis.org

1, 0, -1, -1, 1, 0, -1, -1, 0, 1, -1, 0, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, 0, 0, 0, -1, -1, -1, -1, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, -1, -1, -1, 0, 0, -1, 0, -1, 0, -1, 0, -1, -1, -1, 0, -1, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, -1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -2, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Paul D. Hanna, Jan 06 2015

sign

Compare to the similar series F(x) for the Catalan function C(x) = 1 + x*C(x)^2, where C(F(x)) consists entirely of positive integer coefficients such that C(F(x) - x^k) has negative coefficients for k>0; in which case F(x) = (x+x^2) - (x+x^2)^2, and C(F(x)) = 1/(1-x-x^2).

			G.f.: A(x) = x - x^3 - x^4 + x^5 - x^7 - x^8 + x^10 - x^11 - x^13 - x^14 - x^16 - x^17 - x^18 - x^20 - x^22 - x^26 - x^27 - x^28 - x^29 - x^32 - x^33 - x^35 - x^36 - x^39 - x^41 - x^43 - x^44 - x^45 - x^46 - x^47 - x^48 - x^50 +...
Given the g.f. M(x) of the Motzkin numbers:
M(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 + 323*x^8 + 835*x^9 + 2188*x^10 + 5798*x^11 + 15511*x^12 +...
then
M(A(x)) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 13*x^7 + 19*x^8 + 27*x^9 + 39*x^10 + 55*x^11 + 79*x^12 + 113*x^13 + 160*x^14 +...+ A251571(n)*x^n +...
consists entirely of positive integer coefficients such that M(A(x) - x^k) has negative coefficients for k>0.
Note that a(n) = -2 seems somewhat sparse and occurs at positions:
[58, 123, 181, 187, 203, 213, 230, 236, 245, 253, ...].

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A251571, A251690, A001006.

/* Prints initial N terms: */
N=100;
/* M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of Motzkin numbers: */
{M=1/x*serreverse(x/(1+x+x^2 +x*O(x^(2*N+10))));M +O(x^21) }
/* Print terms as you build vector A, then print A at the end: */
{A=[1, 0]; print1("1, 0, ");
for(l=1, N, A=concat(A, -3);
for(i=1, 4, A[#A]=A[#A]+1;
V=Vec(subst(M, x, x*truncate(Ser(A)) +O(x^floor(2*#A+1)) ));
if((sign(V[2*#A])+1)/2==1, print1(A[#A], ", "); break)); ); A}

A275207 Expansion of (A(x)^2+A(x^2))/2 where A(x) = A001006(x).

Original entry on oeis.org

1, 1, 3, 6, 16, 38, 100, 256, 681, 1805, 4867, 13162, 35925, 98469, 271511, 751656, 2089963, 5831451, 16326785, 45847770, 129108926, 364498596, 1031486590, 2925337352, 8313215743, 23668977163, 67507773621, 192859753310, 551821400008, 1581188102590

Offset: 0

R. J. Mathar, Jul 19 2016

nonn

Analog of A275165 with Motzkin numbers replacing connected graph counts.

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

Cf. A275208, A026940.

b:= proc(n) option remember; `if`(n<2, 1,
      ((3*(n-1))*b(n-2)+(1+2*n)*b(n-1))/(n+2))
    end:
a:= proc(n) option remember; add(b(j)*b(n-j), j=0..n/2)-
      `if`(n::odd, 0, (t-> t*(t-1)/2)(b(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 19 2016

b[n_] := b[n] = If[n<2, 1, ((3*(n-1))*b[n-2] + (1+2*n)*b[n-1])/(n+2)];
a[n_] := a[n] = Sum[b[j]*b[n-j], {j, 0, n/2}] - If[OddQ[n], 0, Function[t, t*(t-1)/2][b[n/2]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 16 2017, after Alois P. Heinz *)

a(2n+1) = A275208(2n+1).
Conjecture: a(2n+1) = A026940(n+1).
Conjecture D-finite with recurrence -3*(n+4)*(n+3)*(29*n-32)*a(n) +10*(29*n-40)*(n+3)*(n+2)*a(n-1) +2*(n+1)*(149*n^2 +208*n-450)*a(n-2) -2*n*(559*n^2 -381*n-1630)*a(n-3) +4*(-68*n^3 +531*n^2 -904*n+351)*a(n-4) +2*(103*n^3-1701*n^2+5330*n -3600)*a(n-5) +18*(11*n^3 -209*n^2 +834*n -778)*a(n-6) +6*n*(269*n-830)*(n-5)*a(n-7) +9*(n-5)*(n-6)*(95*n-134)*a(n-8)=0. - R. J. Mathar, Mar 07 2023
a(n) ~ 3^(n + 5/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

A275208 Expansion of (A(x)^2-A(x^2))/2 where A(x) = A001006(x).

Original entry on oeis.org

0, 1, 2, 6, 14, 38, 96, 256, 672, 1805, 4846, 13162, 35874, 98469, 271384, 751656, 2089640, 5831451, 16325950, 45847770, 129106738, 364498596, 1031480792, 2925337352, 8313200232, 23668977163, 67507731786, 192859753310, 551821286374, 1581188102590

Offset: 0

R. J. Mathar, Jul 19 2016

nonn

Analog of A275166 with Motzkin numbers replacing connected graph counts.

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

Cf. A275207, A026940.

b:= proc(n) option remember; `if`(n<2, 1,
      ((3*(n-1))*b(n-2)+(1+2*n)*b(n-1))/(n+2))
    end:
a:= proc(n) option remember; add(b(j)*b(n-j), j=0..n/2)-
      `if`(n::odd, 0, (t-> t*(t+1)/2)(b(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 19 2016

b[n_] := b[n] = If[n<2, 1, ((3*(n-1))*b[n-2] + (1+2*n)*b[n-1])/(n+2)];
a[n_] := a[n] = Sum[b[j]*b[n-j], {j, 0, n/2}] - If[OddQ[n], 0, Function[t, t*(t + 1)/2][b[n/2]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 16 2017, after Alois P. Heinz *)

a(2n+1) = A275207(2n+1).

cp's OEIS Frontend

A059347 Difference array of Motzkin numbers A001006 read by antidiagonals.

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A116388 Expansion of 1/((1+x*(1-M(x)))*sqrt(1-2*x-3*x^2)), M(x) the g.f. of A001006.

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A121399 G.f. satisfies: A(x) = G(x)*A(x^2*G(x)) where G(x) is the g.f. of the Motzkin numbers (A001006): G = (1 + x*G + x^2*G^2).

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A162482 Expansion of (1/(1-x)^3)*M(x/(1-x)^3), M(x) the g.f. of Motzkin numbers A001006.

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A200538 Product of Jacobsthal and Motzkin numbers: a(n) = A001045(n+1)*A001006(n).

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A200539 Product of Fibonacci and Motzkin numbers: a(n) = A000045(n+1)*A001006(n).

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A200540 Product of Pell and Motzkin numbers: a(n) = A000129(n+1)*A001006(n).

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A251570 G.f. A(x) satisfies the condition that M(A(x)) is a power series in x consisting entirely of positive integer coefficients such that M(A(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + x*M(x) + x*M(x)^2 is the g.f. of the Motzkin numbers A001006.

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A275207 Expansion of (A(x)^2+A(x^2))/2 where A(x) = A001006(x).

A116388 Expansion of 1/((1+x(1-M(x)))sqrt(1-2x-3x^2)), M(x) the g.f. of A001006.

A121399 G.f. satisfies: A(x) = G(x)A(x^2G(x)) where G(x) is the g.f. of the Motzkin numbers (A001006): G = (1 + xG + x^2G^2).

A251570 G.f. A(x) satisfies the condition that M(A(x)) is a power series in x consisting entirely of positive integer coefficients such that M(A(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + xM(x) + xM(x)^2 is the g.f. of the Motzkin numbers A001006.