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 A025177 Triangular array, read by rows: first differences in n,n direction of trinomial array A027907. 23
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS 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 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened FORMULA 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-yz)/[1-z(1+y+y^2)]. EXAMPLE 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 MAPLE 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 MATHEMATICA CoefficientList[CoefficientList[Series[(1 - y*x)/(1 - x*(1 + y + y^2)), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, May 22 2017 *) PROG (PARI) {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)))))}; (PARI) T(n, k)=polcoeff(Ser(polcoeff(Ser((1-y*z)/(1-z*(1+y+y^2)), y), k, y), z), n, z) (PARI) {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)))}; (PARI) 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 CROSSREFS Columns include A025178, A025179, A025180, A025181, A025182. Cf. A024996. Sequence in context: A272896 A188919 A026519 * A026148 A117211 A246576 Adjacent sequences:  A025174 A025175 A025176 * A025178 A025179 A025180 KEYWORD nonn,tabf,easy AUTHOR EXTENSIONS Edited by Ralf Stephan, Jan 09 2005 Offset corrected by R. J. Mathar, Feb 25 2015 STATUS approved

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Last modified October 19 16:17 EDT 2019. Contains 328223 sequences. (Running on oeis4.)