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A083061 Triangle of coefficients of a companion polynomial to the Gandhi polynomial.

%I A083061 #58 Jan 05 2025 19:51:37
%S A083061 1,1,3,4,15,15,34,147,210,105,496,2370,4095,3150,945,11056,56958,
%T A083061 111705,107415,51975,10395,349504,1911000,4114110,4579575,2837835,
%U A083061 945945,135135,14873104,85389132,197722980,244909665,178378200,77567490
%N A083061 Triangle of coefficients of a companion polynomial to the Gandhi polynomial.
%C A083061 This polynomial arises in the setting of a symmetric Bernoulli random walk and occurs in an expression for the even moments of the absolute distance from the origin after an even number of timesteps. The Gandhi polynomial, sequence A036970, occurs in an expression for the odd moments.
%C A083061 When formatted as a square array, first row is A002105, first column is A001147, second column is A001880.
%C A083061 Another version of the triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 3, 6, 10, 15, 21, 28, ...] DELTA [1, 2, 3, 4, 5, 6, 7, 8, 9, ...] = 1; 0, 1; 0, 1, 3; 0, 4, 15, 15; 0, 34, 147, 210, 105; ... where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jun 07 2004
%C A083061 In A160464 we defined the coefficients of the ES1 matrix. Our discovery that the n-th term of the row coefficients ES1[1-2*m,n] for m>=1, can be generated with rather simple polynomials led to triangle A094665 and subsequently to this one. - _Johannes W. Meijer_, May 24 2009
%C A083061 Related to polynomials defined in A160485 by a shift of +-1/2 and scaling by a power of 2. - _Richard P. Brent_, Jul 15 2014
%H A083061 R. P. Brent, <a href="http://arxiv.org/abs/1407.3533">Generalising Tuenter's binomial sums</a>, arXiv:1407.3533 [math.CO], 2014.
%H A083061 R. B. Brent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Brent/brent5.html">Generalizing Tuenter's Binomial Sums</a>, J. Int. Seq. 18 (2015) # 15.3.2.
%H A083061 Marc Joye, Pascal Paillier and Berry Schoenmakers, <a href="http://www.win.tue.nl/~berry/papers/ches05hodpa.pdf">On Second-Order Differential Power Analysis</a>, in Cryptographic Hardware and Embedded Systems-CHES 2005, editors: Josyula R. Rao and Berk Sunar, Lecture Notes in Computer Science 3659 (2005) 293-308, Springer-Verlag.
%H A083061 H. J. H. Tuenter, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/40-2/tuenter.pdf">Walking into an absolute sum</a>, The Fibonacci Quarterly, 40 (2002), 175-180.
%F A083061 Let T(i, x)=(2x+1)(x+1)T(i-1, x+1)-2x^2T(i-1, x), T(0, x)=1; so that T(1, x)=1+3x; T(2, x)=4+15x+15x^2; T(3, x)=34+147x+210x^2+105x^3, etc. Then the (i, j)-th entry in the table is the coefficient of x^j in T(i, x).
%F A083061 a(n, k)*2^(n-k) = A085734(n, k). - _Philippe Deléham_, Feb 27 2005
%e A083061 Triangle starts (with an additional first column 1,0,0,...):
%e A083061 [1]
%e A083061 [0,      1]
%e A083061 [0,      1,       3]
%e A083061 [0,      4,      15,      15]
%e A083061 [0,     34,     147,     210,     105]
%e A083061 [0,    496,    2370,    4095,    3150,     945]
%e A083061 [0,  11056,   56958,  111705,  107415,   51975,  10395]
%e A083061 [0, 349504, 1911000, 4114110, 4579575, 2837835, 945945, 135135]
%p A083061 imax := 6;
%p A083061 T1(0, x) := 1:
%p A083061 T1(0, x+1) := 1:
%p A083061 for i from 1 to imax do
%p A083061     T1(i, x) := expand((2*x+1) * (x+1) * T1(i-1, x+1) - 2*x^2*T1(i-1, x)):
%p A083061     dx := degree(T1(i, x)):
%p A083061     for k from 0 to dx do
%p A083061         c(k) := coeff(T1(i, x), x, k)
%p A083061     od:
%p A083061     T1(i, x+1) := sum(c(j1)*(x+1)^(j1), j1 = 0..dx):
%p A083061 od:
%p A083061 for i from 0 to imax do
%p A083061     for j from 0 to i do
%p A083061         a(i, j) := coeff(T1(i, x), x, j)
%p A083061     od:
%p A083061 od:
%p A083061 seq(seq(a(i, j), j = 0..i), i = 0..imax);
%p A083061 # _Johannes W. Meijer_, Jun 27 2009, revised Sep 23 2012
%t A083061 b[0, 0] = 1;
%t A083061 b[n_, k_] := b[n, k] = Sum[2^j*(Binomial[k + j, 1 + j] + Binomial[k + j + 1, 1 + j])*b[n - 1, k - 1 + j], {j, Max[0, 1 - k], n - k}];
%t A083061 a[0, 0] = 1;
%t A083061 a[n_, k_] := b[n, k]/2^(n - k);
%t A083061 Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 19 2018, after _Philippe Deléham_ *)
%o A083061 (Sage) # uses[fr2_row from A088874]
%o A083061 A083061_row = lambda n: [(-1)^(n-k)*m*2^(-n+k) for k,m in enumerate(fr2_row(n))]
%o A083061 for n in (0..7): print(A083061_row(n)) # _Peter Luschny_, Sep 19 2017
%Y A083061 Cf. A036970, A160485.
%Y A083061 From _Johannes W. Meijer_, May 24 2009 and Jun 27 2009: (Start)
%Y A083061 Cf. A160464, A094665 and A160468.
%Y A083061 A002105 equals the row sums (n>=2) and the first left hand column (n>=1).
%Y A083061 A001147, A001880, A160470, A160471 and A160472 are the first five right hand columns.
%Y A083061 Appears in A162005, A162006 and A162007.
%Y A083061 (End)
%K A083061 nonn,tabl
%O A083061 0,3
%A A083061 _Hans J. H. Tuenter_, Apr 19 2003