This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A052509 Knights-move Pascal triangle: T(n,k), n >= 0, 0 <= k <= n; T(n,0) = T(n,n) = 1, T(n,k) = T(n-1,k) + T(n-2,k-1) for k = 1,2,...,n-1, n >= 2. 17
 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 4, 2, 1, 1, 6, 11, 8, 4, 2, 1, 1, 7, 16, 15, 8, 4, 2, 1, 1, 8, 22, 26, 16, 8, 4, 2, 1, 1, 9, 29, 42, 31, 16, 8, 4, 2, 1, 1, 10, 37, 64, 57, 32, 16, 8, 4, 2, 1, 1, 11, 46, 93, 99, 63, 32, 16, 8, 4, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also square array T(n,k) (n >= 0, k >= 0) read by antidiagonals: T(n,k) = Sum_{i=0..k} binomial(n,i). As a number triangle read by rows, this is T(n,k) = Sum_{i=n-2*k..n-k} binomial(n-k,i), with T(n,k) = T(n-1,k) + T(n-2,k-1). Row sums are A000071(n+2). Diagonal sums are A023435(n+1). It is the reverse of the Whitney triangle A004070. - Paul Barry, Sep 04 2005 Also, twice number of orthants intersected by a generic k-dimensional subspace of R^n [Naiman and Scheinerman, 2017]. - N. J. A. Sloane, Mar 03 2018 LINKS Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened C. Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 1C. Daniel Q. Naiman, Edward R. Scheinerman, Arbitrage and Geometry, arXiv:1709.07446 [q-fin.MF], 2017 [Contains the square array multiplied by 2]. Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See Tables 5 and 14. D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150. FORMULA T(n, k) = Sum_{m=0..n} binomial(n-k, k-m). - Wouter Meeussen, Oct 03 2002 From Werner Schulte, Feb 15 2018: (Start) Referring to the square array T(i,j): G.f. of row n: Sum_{k>=0} T(n,k) * x^k = (1+x)^n / (1-x). G.f. of T(i,j): Sum_{k>=0, n>=0} T(n,k) * x^k * y^n = 1 / ((1-x)*(1-y-x*y). Let a_i(n) be multiplicative with a_i(p^e) = T(i, e), p prime and e >= 0, then Sum_{n>0} a_i(n)/n^s = (zeta(s))^(i+1) / (zeta(2*s))^i for i >= 0. (End) EXAMPLE Triangle begins:   1   1,  1   1,  2,  1   1,  3,  2,  1   1,  4,  4,  2,  1   1,  5,  7,  4,  2,  1   1,  6, 11,  8,  4,  2,  1 As a square array, this begins:   1  1  1  1  1  1 ...   1  2  2  2  2  2 ...   1  3  4  4  4  4 ...   1  4  7  8  8  8 ...   1  5 11 15 16 ...   1  6 16 26 31 32 ... MAPLE a := proc(n::nonnegint, k::nonnegint) option remember: if k=0 then RETURN(1) fi: if k=n then RETURN(1) fi: a(n-1, k)+a(n-2, k-1) end: for n from 0 to 11 do for k from 0 to n do printf(`%d, `, a(n, k)) od: od: # James A. Sellers, Mar 17 2000 with(combinat): for s from 0 to 11 do for n from s to 0 by -1 do if n=0 or s-n=0 then printf(`%d, `, 1) else printf(`%d, `, sum(binomial(n, i), i=0..s-n)) fi; od: od: # James A. Sellers, Mar 17 2000 MATHEMATICA Table[Sum[Binomial[n-k, k-m], {m, 0, n}], {n, 0, 10}, {k, 0, n}] PROG (PARI) T(n, k)=sum(m=0, n, binomial(n-k, k-m)); for(n=0, 10, for(k=0, n, print1(T(n, k), ", "); ); print(); ); /* show triangle */ (Haskell) a052509 n k = a052509_tabl !! n !! k a052509_row n = a052509_tabl !! n a052509_tabl = [1] : [1, 1] : f [1] [1, 1] where    f row' row = rs : f row rs where      rs = zipWith (+) ([0] ++ row' ++ [1]) (row ++ [0]) -- Reinhard Zumkeller, Nov 22 2012 (GAP) A052509:=Flat(List([0..100], n->List([0..n], k->Sum([0..n], m->Binomial(n-k, k-m))))); # Muniru A Asiru, Sat Feb 17 2018 (MAGMA) [[(&+[Binomial(n-k, k-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 13 2019 (Sage) [[sum(binomial(n-k, k-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 13 2019 CROSSREFS Cf. A054123, A054124, A007318, A008949. Row sums A000071; Diagonal sums A023435; Mirror A004070. Columns give A000027, A000124, A000125, A000127, A006261, ... Cf. A052509, A054123, A054124, A007318, A008949, A052553. Partial sums across rows of (extended) Pascal's triangle A052553. Sequence in context: A194005 A055794 A092905 * A172119 A228125 A227588 Adjacent sequences:  A052506 A052507 A052508 * A052510 A052511 A052512 KEYWORD nonn,tabl,easy,nice AUTHOR N. J. A. Sloane, Mar 17 2000 EXTENSIONS More terms from James A. Sellers, Mar 17 2000 Entry formed by merging two earlier entries. - N. J. A. Sloane, Jun 17 2007 Edited by Johannes W. Meijer, Jul 24 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 15 10:15 EDT 2019. Contains 328026 sequences. (Running on oeis4.)