A378576 G.f. A(x) satisfies A(x) = x + x*A(A(A(A(A(A(x)))))), so that this sequence shifts left under the 6th self-COMPOSE.

1, 1, 6, 66, 981, 17576, 359101, 8109026, 198480901, 5197916551, 144326504376, 4220683214771, 129349023338616, 4138098976882836, 137770738524681831, 4761510500867829696, 170476220596490911691, 6311806909067054474161, 241301669599996597349506, 9512867537981387958217696, 386276838889195561353811321

Offset

1,3

Comments

Conjecture: a(n) == 1 (mod 5) for n >= 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..400

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas, wherein A^n(x) denotes the n-th iteration of A(x) with A^0(x) = x.

(1) A(x) = x + x*A^6(x).

(2) A(x) = A(A(x))/(1 + A^7(x)).

(3) A(x) = Series_Reversion( x/(1 + A^5(x)) ).

(4) A(x) = Sum_{n>=0} Product_{k=0..n} A^(5*k)(x).

(5) A^n(x) = A^(n+1)(x) / (1 + A^(n+6)(x)) for n >= 0.

(6) A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+6)(x)) for n >= 1.

From Seiichi Manyama, Jun 13 2026: (Start)

G.f. A(x) satisfies A(x) = x*(1 + A^l(x)), where A^l(x) denotes the l-th iterate of A.

Let a(n,k,l) = [x^n] A^k(x), where A^k(x) is the k-th iterate of A.

a(n,0,l) = 0^(n-1) and a(n,k,l) = a(n,k-1,l) + Sum_{j=1..n-1} a(j,k+l-1,l) * a(n-j,k-1,l) for k > 0. (End)

Example

G.f.: A(x) = x + x^2 + 6*x^3 + 66*x^4 + 981*x^5 + 17576*x^6 + 359101*x^7 + 8109026*x^8 + 198480901*x^9 + 5197916551*x^10 + ...
where A(x) = x + x*A^6(x).
RELATED SERIES.
  A^2(x) = x + 2*x^2 + 14*x^3 + 163*x^4 +  2496*x^5 +  45577*x^6 +  944034*x^7 + ...
  A^3(x) = x + 3*x^2 + 24*x^3 + 297*x^4 +  4711*x^5 +  88073*x^6 + 1856179*x^7 + ...
  A^4(x) = x + 4*x^2 + 36*x^3 + 474*x^4 +  7816*x^5 + 150144*x^6 + 3230016*x^7 + ...
  A^5(x) = x + 5*x^2 + 50*x^3 + 700*x^4 + 12025*x^5 + 238000*x^6 + 5240145*x^7 + ...
  A^6(x) = x + 6*x^2 + 66*x^3 + 981*x^4 + 17576*x^5 + 359101*x^6 + 8109026*x^7 + ...
  ...
By formula (4),
A(x) = x + x*A^5(x) + x*A^5(x)*A^10(x) + x*A^5(x)*A^10(x)*A^15(x) + x*A^5(x)*A^10(x)*A^15(x)*A^20(x) + ...
Examples of formula (5), A^n(x) = A^(n+1)(x)/(1 + A^(n+6)(x)):
  n=0: x = A(x)/(1 + A(A(A(A(A(A(x))))))),
  n=1: A(x) = A(A(x))/(1 + A(A(A(A(A(A(A(x)))))))),
  n=2: A(A(x)) = A(A(A(x)))/(1 + A(A(A(A(A(A(A(A(x))))))))),
  n=3: A(A(A(x))) = A(A(A(A(x))))/(1 + A(A(A(A(A(A(A(A(A(x)))))))))),
  ...
Examples of formula (6), A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+6)(x)):
  n=1: A(x) = x*(1 + A(A(A(A(A(A(x))))))),
  n=2: A(A(x)) = x*(1 + A(A(A(A(A(A(x)))))))*(1 + A(A(A(A(A(A(A(x)))))))),
  n=3: A(A(A(x))) = x*(1 + A(A(A(A(A(A(x)))))))*(1 + A(A(A(A(A(A(A(x))))))))*(1 + A(A(A(A(A(A(A(A(x))))))))),
  ...

Prog

(PARI) /* By definition, A(x) = x + x*A(A(A(A(A(A(x)))))) */
/* Define the n-th iteration of function F: */
{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
{a(n) = my(A=x); for(i=1, n, A = x + x*ITERATE(6, A, n)); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI)
a_vector(n, k=1, l=6) = {
  my(k_limit(row)=k+(n-row)*(l-1), A=vector(n, row, vector(k_limit(row)+1)));
  for(col=0, k_limit(1), A[1][col+1]=1);
  for(row=2, n, A[row][1]=0);
  for(row=2, n,
    for(col=1, k_limit(row),
      A[row][col+1]=A[row][col]+sum(j=1, row-1, A[j][col+l]*A[row-j][col]);
    );
  );
  vector(n, row, A[row][k+1])
}; \\ Seiichi Manyama, Jun 13 2026

Crossrefs

Cf. A030266, A091713, A196523, A378575.

Sequence in context: A396281 A371681 A216636 * A169715 A211824 A375950

Adjacent sequences: A378573 A378574 A378575 * A378577 A378578 A378579

Keyword

nonn

Author

Paul D. Hanna, Dec 01 2024

Status

approved