A054335 A convolution triangle of numbers based on A000984 (central binomial coefficients of even order).
1, 2, 1, 6, 4, 1, 20, 16, 6, 1, 70, 64, 30, 8, 1, 252, 256, 140, 48, 10, 1, 924, 1024, 630, 256, 70, 12, 1, 3432, 4096, 2772, 1280, 420, 96, 14, 1, 12870, 16384, 12012, 6144, 2310, 640, 126, 16, 1, 48620, 65536, 51480, 28672, 12012, 3840, 924, 160, 18, 1
Offset: 0
Examples
Triangle begins:
1;
2, 1;
6, 4, 1;
20, 16, 6, 1;
70, 64, 30, 8, 1;
252, 256, 140, 48, 10, 1;
924, 1024, 630, 256, 70, 12, 1; ...
Fourth row polynomial (n=3): p(3,x) = 20 + 16*x + 6*x^2 + x^3.
From _Paul Barry_, May 06 2009: (Start)
Production matrix begins
2, 1;
2, 2, 1;
0, 2, 2, 1;
-2, 0, 2, 2, 1;
0, -2, 0, 2, 2, 1;
4, 0, -2, 0, 2, 2, 1;
0, 4, 0, -2, 0, 2, 2, 1;
-10, 0, 4, 0, -2, 0, 2, 2, 1;
0, -10, 0, 4, 0, -2, 0, 2, 2, 1; (End)
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
- Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
- Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Programs
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GAP
T:= function(n, k) if k mod 2=0 then return Binomial(2*n-k, n-Int(k/2))*Binomial(n-Int(k/2),Int(k/2))/Binomial(k,Int(k/2)); else return 4^(n-k)*Binomial(n-Int((k-1)/2)-1, Int((k-1)/2)); fi; end; Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Jul 20 2019 -
Magma
T:= func< n, k | (k mod 2) eq 0 select Binomial(2*n-k, n-Floor(k/2))* Binomial(n-Floor(k/2),Floor(k/2))/Binomial(k,Floor(k/2)) else 4^(n-k)*Binomial(n-Floor((k-1)/2)-1, Floor((k-1)/2)) >; [[T(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jul 20 2019
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Maple
A054335 := proc(n,k) if k <0 or k > n then 0 ; elif type(k,odd) then kprime := floor(k/2) ; binomial(n-kprime-1,kprime)*4^(n-k) ; else kprime := k/2 ; binomial(2*n-k,n-kprime)*binomial(n-kprime,kprime)/binomial(k,kprime) ; end if; end proc: # R. J. Mathar, Mar 12 2013 # Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left. PMatrix(10, n -> binomial(2*(n-1), n-1)); # Peter Luschny, Oct 19 2022
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Mathematica
Flatten[ CoefficientList[#1, x] & /@ CoefficientList[ Series[1/(Sqrt[1 - 4*z] - x*z), {z, 0, 9}], z]] (* or *) a[n_, k_?OddQ] := 4^(n-k)*Binomial[(2*n-k-1)/2, (k-1)/2]; a[n_, k_?EvenQ] := (Binomial[n-k/2, k/2]*Binomial[2*n-k, n-k/2])/Binomial[k, k/2]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 08 2011, updated Jan 16 2014 *) -
PARI
T(n, k) = if(k%2==0, binomial(2*n-k, n-k/2)*binomial(n-k/2,k/2)/binomial(k,k/2), 4^(n-k)*binomial(n-(k-1)/2-1, (k-1)/2)); for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 20 2019
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Sage
def T(n, k): if (mod(k,2)==0): return binomial(2*n-k, n-k/2)*binomial(n-k/2,k/2)/binomial(k,k/2) else: return 4^(n-k)*binomial(n-(k-1)/2-1, (k-1)/2) [[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 20 2019
Formula
a(n, 2*k+1) = binomial(n-k-1, k)*4^(n-2*k-1), a(n, 2*k) = binomial(2*(n-k), n-k)*binomial(n-k, k)/binomial(2*k, k), k >= 0, n >= m >= 0; a(n, m) := 0 if n
Column recursion: a(n, m)=2*(2*n-m-1)*a(n-1, m)/(n-m), n>m >= 0, a(m, m) := 1.
G.f. for column m: cbie(x)*(x*cbie(x))^m, with cbie(x) := 1/sqrt(1-4*x).
G.f.: 1/(1-x*y-2*x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction). - Paul Barry, May 06 2009
Sum_{k=0..floor(n/2)} T(n-k,n-2*k) = A098615(n). - Philippe Deléham, Feb 01 2012
T(n,k) = 4*T(n-1,k) + T(n-2,k-2) for k>=1. - Philippe Deléham, Feb 02 2012
Vertical recurrence: T(n,k) = 1*T(n-1,k-1) + 2*T(n-2,k-1) + 6*T(n-3,k-1) + 20*T(n-4,k-1) + ... for k >= 1 (the coefficients 1, 2, 6, 20, ... are the central binomial coefficients A000984). - Peter Bala, Oct 17 2015
Comments