0,3

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 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 07 2004

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009: (Start)

In A160464 we defined the coefficients of the ES1 matrix. Our discovery that the nth term of the row coefficients ES1[1-2*m,n] for m = 1, 2, 3, .. , can be generated with rather simple polynomials led to triangle A094665 and subsequently to this one.

(End)

Hans J. H. Tuenter, "Walking into an absolute sum", The Fibonacci Quarterly, 40(2):175-180, 2002.

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.

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) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 27 2005

1; 1,3; 4,15,15; 34,147,210,105; 496,2370,4095,3150,945; ...

restart; imax:=7; i:=0: 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(j)*(x+1)^(j), j=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: T:=0; for i from 0 to imax do for j from 0 to i do a(T):= a(i, j); T:=T+1; od: od: seq(a(n), n=0..T-1); [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 2009]

Cf. A036970.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 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.

(End)

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 2009: (Start)

Appears in A162005, A162006 and A162007.

(End)

Sequence in context: A130113 A004735 A066830 this_sequence A136641 A053359 A056742

Adjacent sequences: A083058 A083059 A083060 this_sequence A083062 A083063 A083064

nonn,tabl

Hans J. H. Tuenter (HTuenter(AT)schulich.yorku.ca), Apr 19 2003