A378575 G.f. A(x) satisfies A(x) = x + x*A(A(A(A(A(x))))), so that this sequence shifts left under the 5th self-COMPOSE.

1, 1, 5, 45, 545, 7945, 132005, 2423501, 48224129, 1026722489, 23177970949, 551133715197, 13734995332769, 357361170997321, 9677345660994725, 272075021315860781, 7925076713952829697, 238747406787319312025, 7427421640015549840133, 238301672444134819413533, 7875799810817511976148129

Offset

1,3

Comments

Conjecture: a(n) == 1 (mod 4) 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^5(x).

(2) A(x) = A(A(x))/(1 + A^6(x)).

(3) A(x) = Series_Reversion( x/(1 + A^4(x)) ).

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

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

(6) A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+5)(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 + 5*x^3 + 45*x^4 + 545*x^5 + 7945*x^6 + 132005*x^7 + 2423501*x^8 + 48224129*x^9 + 1026722489*x^10 + ...
where A(x) = x + x*A^5(x).
RELATED SERIES.
  A^2(x) = x + 2*x^2 + 12*x^3 + 116*x^4 + 1460*x^5 +  21820*x^6 +  369140*x^7 +  6873732*x^8 + ...
  A^3(x) = x + 3*x^2 + 21*x^3 + 219*x^4 + 2885*x^5 +  44483*x^6 +  770269*x^7 + 14610939*x^8 + ...
  A^4(x) = x + 4*x^2 + 32*x^3 + 360*x^4 + 4984*x^5 +  79648*x^6 + 1417768*x^7 + 27500512*x^8 + ...
  A^5(x) = x + 5*x^2 + 45*x^3 + 545*x^4 + 7945*x^5 + 132005*x^6 + 2423501*x^7 + 48224129*x^8 + ...
  ...
By formula (4),
A(x) = x + x*A^4(x) + x*A^4(x)*A^8(x) + x*A^4(x)*A^8(x)*A^12(x) + x*A^4(x)*A^8(x)*A^12(x)*A^16(x) + ...
Examples of formula (5), A^n(x) = A^(n+1)(x)/(1 + A^(n+5)(x)):
  n=0: x = A(x)/(1 + A(A(A(A(A(x)))))),
  n=1: A(x) = A(A(x))/(1 + A(A(A(A(A(A(x))))))),
  n=2: A(A(x)) = A(A(A(x)))/(1 + 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(x))))))))),
  ...
Examples of formula (6), A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+5)(x)):
  n=1: A(x) = x*(1 + A(A(A(A(A(x)))))),
  n=2: A(A(x)) = x*(1 + A(A(A(A(A(x))))))*(1 + A(A(A(A(A(A(x))))))),
  n=3: A(A(A(x))) = x*(1 + A(A(A(A(A(x))))))*(1 + A(A(A(A(A(A(x)))))))*(1 + A(A(A(A(A(A(A(x)))))))),
  ...

Prog

(PARI) /* By definition, A(x) = x + x*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(5, A, n)); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI)
a_vector(n, k=1, l=5) = {
  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.

Cf. A395269。

Sequence in context: A189122 A062023 A169714 * A084095 A375952 A375951

Adjacent sequences: A378572 A378573 A378574 * A378576 A378577 A378578

Keyword

nonn,changed

Author

Paul D. Hanna, Dec 01 2024

Status

approved