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

1, 1, 3, 4, 15, 15, 34, 147, 210, 105, 496, 2370, 4095, 3150, 945, 11056, 56958, 111705, 107415, 51975, 10395, 349504, 1911000, 4114110, 4579575, 2837835, 945945, 135135, 14873104, 85389132, 197722980, 244909665, 178378200, 77567490

Offset: 0

Author

Hans J. H. Tuenter, Apr 19 2003

Keywords

• nonn
• tabl

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

Examples

			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]
		

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.

Crossrefs

Cf. A036970, A160485.

From Johannes W. Meijer, May 24 2009 and Jun 27 2009: (Start)

Cf. A160464, A094665 and A160468.

A002105 equals the row sums (n>=2) and the first left hand column (n>=1).

A001147, A001880, A160470, A160471 and A160472 are the first five right hand columns.

Appears in A162005, A162006 and A162007.

(End)

Programs

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 *)

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

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