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.
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
Offset: 0
Examples
Triangle begins: [0] 1; [1] 1, 1; [2] 1, 2, 1; [3] 1, 3, 2, 1; [4] 1, 4, 4, 2, 1; [5] 1, 5, 7, 4, 2, 1; [6] 1, 6, 11, 8, 4, 2, 1; [7] 1, 7, 16, 15, 8, 4, 2, 1; [8] 1, 8, 22, 26, 16, 8, 4, 2, 1; [9] 1, 9, 29, 42, 31, 16, 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 ...
Links
- Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
- Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 1C.
- Daniel Q. Naiman and 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.
- Index entries for triangles and arrays related to Pascal's triangle
Crossrefs
Programs
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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
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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 -
Magma
[[(&+[Binomial(n-k, k-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 13 2019
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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 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 Sellers, Mar 17 2000
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Mathematica
Table[Sum[Binomial[n-k, k-m], {m, 0, n}], {n, 0, 10}, {k, 0, n}] T[n_, k_] := Hypergeometric2F1[-k, -n + k, -k, -1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Nov 28 2021 *) -
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 */
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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
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)
T(n, k) = hypergeom([-k, -n + k], [-k], -1). - Peter Luschny, Nov 28 2021
From Jianing Song, May 30 2022: (Start)
Referring to the triangle, G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = 1 / ((1-x*y)*(1-x-x^2*y)).
T(n,k) = 2^(n-k) for ceiling(n/2) <= k <= n. (End)
Extensions
More terms from James 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
Comments