A177897 Triangle of octanomial coefficients read by rows: n-th row is obtained by expanding ((1+x)*(1+x^2)*(1+x^4))^n mod 2 and converting to decimal.

1, 255, 21845, 3342387, 286331153, 64424509455, 5519032976645, 844437815230467, 72340172838076673, 18446744073709551615, 1567973246265311887445, 241781474574111093044019, 20552052528097949033496593, 4660480146812799619066433295, 396140812663555408357742346245, 61084913312720243968763869790979

Offset: 0

Author

Vladimir Shevelev, Dec 15 2010

Keywords

• nonn
• easy

Comments

A generalization: Denote {a_k(n)}_(n>=0) the sequence of triangle of 2^k-nomial coefficients [read by rows: n-th row is obtained by expanding ((1+x)*(1+x^2)*...*(1+x^(2^(k-1)))^n ] mod 2 converted to decimal. Then a_k(n)=A001317((2^k-1)*n). [Proof is based on the fact (following from the Lucas theorem for the binomial coefficients) that the k-th row of Pascal triangle contains odd coefficients only iff k is Mersenne number (k=2^m-1)].

Crossrefs

Cf. A001317, A008589, A177882.

Programs

Mathematica

a = Plus@@(x^Range[0, 7]); Table[FromDigits[Mod[CoefficientList[a^n, x], 2], 2], {n, 0, 15}]

PARI

a(n) = subst(lift(Mod(1+'x, 2)^(7*n)), 'x, 2); \\ Michel Marcus, Oct 14 2024

Python

def A177897(n): return sum((bool(~(m:=7*n)&m-k)^1)<Chai Wah Wu, May 03 2023

Formula

a(n) = A001317(7*n).

Extensions

More terms from Michel Marcus, Oct 14 2024