A025182 -id:A025182 - cp's OEIS frontend

A025177 Triangular array, read by rows: first differences in n,n direction of trinomial array A027907.

1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 2, 4, 4, 4, 2, 1, 1, 3, 7, 10, 12, 10, 7, 3, 1, 1, 4, 11, 20, 29, 32, 29, 20, 11, 4, 1, 1, 5, 16, 35, 60, 81, 90, 81, 60, 35, 16, 5, 1, 1, 6, 22, 56, 111, 176, 231, 252, 231, 176, 111, 56, 22, 6, 1, 1, 7, 29, 84, 189, 343, 518, 659, 714, 659, 518, 343

Offset: 0

Clark Kimberling

The Motzkin transforms of the rows starting (1, 2), (1, 3) and (1, 4), extended by zeros after their last element, are apparently in A026134, A026109 and A026110. - R. J. Mathar, Dec 11 2008

			               1
            1  0  1
         1  1  2  1  1
      1  2  4  4  4  2  1
   1  3  7 10 12 10  7  3  1
1  4 11 20 29 32 29 20 11  4  1

Georg Fischer, Table of n, a(n) for n = 0..2499 [rows 0..49; first 676 terms from _G. C. Greubel_]

Columns include A025178, A025179, A025180, A025181, A025182.

Cf. A024996, A025192 (row sums).

A025177 := proc(n,k)
    option remember;
    if k < 0 or k > 2*n then
        0;
    elif n = 0 then
        1 ;
    elif n = 1 then
        op(k+1,[1,0,1]) ;
    else
        procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc:
seq(seq(A025177(n,k),k=0..2*n),n=0..20)  ; # R. J. Mathar, Feb 25 2015

nmax = 10; CoefficientList[CoefficientList[Series[(1 - y*x)/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* G. C. Greubel, May 22 2017; amended by Georg Fischer, Jun 24 2020 *)

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1,0,1][k+1], if( n==2, [1,1,2,1,1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};

T(n,k)=polcoeff(Ser(polcoeff(Ser((1-y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, polcoeff( (1 + x + x^2)^n, k) - polcoeff( (1 + x + x^2)^(n-1), k-1)))};

g=matrix(33,65);
for(n=0,32,for(k=0,2*n,g[n+1,k+1]=0));
g[1,1]=1;
g[2,1]=1;g[2,2]=0;g[2,3]=1;
g[3,1]=1;g[3,2]=1;g[3,3]=2;g[3,4]=1;g[3,5]=1;
for(n=0,2,for(k=0,2*n,print(n," ",k," ",g[n+1,k+1])))
for(n=3,32,g[n+1,1]=1;print(n," 1 1");g[n+1,2]=n-1;print(n," 2 ",n-1);for(k=2,2*n,g[n+1,k+1]=g[n,k-1]+g[n,k]+g[n,k+1];print(n," ",k," ",g[n+1,k+1])))
\\ Michael B. Porter, Feb 02 2010

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1].

G.f.: (1-y*z)/[1-z*(1+y+y^2)].

Edited by Ralf Stephan, Jan 09 2005

Offset corrected by R. J. Mathar, Feb 25 2015

A014533 Form array in which n-th row is obtained by expanding (1 + x + x^2)^n and taking the 4th column from the center.

Original entry on oeis.org

1, 5, 21, 77, 266, 882, 2850, 9042, 28314, 87802, 270270, 827190, 2520336, 7651632, 23162976, 69954048, 210859245, 634569201, 1907165337, 5725520801, 17172595110, 51465297950, 154135675070, 461366154990, 1380317174145

Offset: 1

N. J. A. Sloane

First differences seem to be in A025182.
a(n-3) = A111808(n, n-4) for n > 3. - Reinhard Zumkeller, Aug 17 2005
a(n-4) = number of paths in the half-plane x >= 0, from (0,0) to (n,4), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=5, we have the 5 paths HUUUU, UHUUU, UUHUU, UUUHU, UUUUH. - José Luis Ramírez Ramírez, Apr 19 2015

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Trinomial Coefficient

a := n -> simplify(GegenbauerC(n-1, -n-3, -1/2)):
seq(a(n), n=1..25); # Peter Luschny, May 09 2016

Rest[CoefficientList[Series[x*((1-x-Sqrt[1-2*x-3*x^2])/(2*x^2))^4/(1-x-2*x^2*(1-x-Sqrt[1-2*x-3*x^2])/(2*x^2)), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 20 2015 *)
Table[GegenbauerC[n-1, -n - 3, -1/2], {n,0,50}] (* G. C. Greubel, Feb 28 2017 *)

x='x + O('x^50); Vec(x*((1-x-sqrt(1-2*x-3*x^2))/(2*x^2))^4/(1-x-2*x^2*(1-x-sqrt(1-2*x-3*x^2))/(2*x^2))) \\ G. C. Greubel, Feb 28 2017

Conjecture: -(n+7)*(n-1)*a(n) + (n+3)*(2*n+5)*a(n-1) + 3*(n+3)*(n+2)*a(n-2) = 0. - R. J. Mathar, Feb 25 2015
G.f.: z*M(z)^4/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) ~ 3^(n+7/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 20 2015
From Peter Luschny, May 09 2016: (Start)
a(n) = C(6+2*n, n-1)*hypergeom([-n+1, -n-7], [-5/2-n], 1/4).
a(n) = GegenbauerC(n-1, -n-3, -1/2). (End)

More terms from James Sellers, Feb 05 2000

A026071 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 4. Also a(n) = T(n,n-4), where T is the array defined in A024996.

Original entry on oeis.org

1, 3, 12, 40, 133, 427, 1352, 4224, 13080, 40216, 122980, 374452, 1136226, 3438150, 10380048, 31279728, 94114125, 282804759, 848886180, 2545759328, 7628718845, 22845628531, 68377674280, 204560102800, 611720539235, 1828673918721

Offset: 4

Clark Kimberling

nonn

First differences of A025182.

Conjecture: -(n-4)*(n+4)*a(n) +(4*n+7)*(n-4)*a(n-1) +(-2*n^2+23*n-12)*a(n-2) -(4*n+3)*(n-4)*a(n-3) +3*(n-4)*(n-5)*a(n-4)=0. - R. J. Mathar, Jun 22 2013

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A025177 Triangular array, read by rows: first differences in n,n direction of trinomial array A027907.

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A014533 Form array in which n-th row is obtained by expanding (1 + x + x^2)^n and taking the 4th column from the center.

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A026071 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 4. Also a(n) = T(n,n-4), where T is the array defined in A024996.

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