OFFSET
0,3
COMMENTS
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.
When formatted as a square array, first row is A002105, first column is A001147, second column is A001880.
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
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
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
LINKS
R. P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
Marc Joye, Pascal Paillier and Berry Schoenmakers, On Second-Order Differential Power Analysis, 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. J. H. Tuenter, Walking into an absolute sum, The Fibonacci Quarterly, 40 (2002), 175-180.
FORMULA
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).
a(n, k)*2^(n-k) = A085734(n, k). - Philippe Deléham, Feb 27 2005
EXAMPLE
Triangle starts (with an additional first column 1,0,0,...):
[1]
[0, 1]
[0, 1, 3]
[0, 4, 15, 15]
[0, 34, 147, 210, 105]
[0, 496, 2370, 4095, 3150, 945]
[0, 11056, 56958, 111705, 107415, 51975, 10395]
[0, 349504, 1911000, 4114110, 4579575, 2837835, 945945, 135135]
MAPLE
imax := 6;
T1(0, x) := 1:
T1(0, x+1) := 1:
for i from 1 to imax do
T1(i, x) := expand((2*x+1) * (x+1) * T1(i-1, x+1) - 2*x^2*T1(i-1, x)):
dx := degree(T1(i, x)):
for k from 0 to dx do
c(k) := coeff(T1(i, x), x, k)
od:
T1(i, x+1) := sum(c(j1)*(x+1)^(j1), j1 = 0..dx):
od:
for i from 0 to imax do
for j from 0 to i do
a(i, j) := coeff(T1(i, x), x, j)
od:
od:
seq(seq(a(i, j), j = 0..i), i = 0..imax);
# Johannes W. Meijer, Jun 27 2009, revised Sep 23 2012
MATHEMATICA
b[0, 0] = 1;
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}];
a[0, 0] = 1;
a[n_, k_] := b[n, k]/2^(n - k);
Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 19 2018, after Philippe Deléham *)
PROG
(Sage) # uses[fr2_row from A088874]
A083061_row = lambda n: [(-1)^(n-k)*m*2^(-n+k) for k, m in enumerate(fr2_row(n))]
for n in (0..7): print(A083061_row(n)) # Peter Luschny, Sep 19 2017
CROSSREFS
From Johannes W. Meijer, May 24 2009 and Jun 27 2009: (Start)
A002105 equals the row sums (n>=2) and the first left hand column (n>=1).
(End)
KEYWORD
nonn,tabl
AUTHOR
Hans J. H. Tuenter, Apr 19 2003
STATUS
approved