A309509 -id:A309509 - cp's OEIS frontend

A372492 G.f. satisfies A(A(A(A(x)))) = F(x), where F(x) is the g.f. for A002697(n) = n*4^(n-1).

0, 1, 2, 0, 4, -8, -8, 288, -1712, -1888, 105472, -288576, -10404800, 84940672, 1454871936, -24372060160, -255228956416, 8232158755328, 49829958005760, -3390379506089984, -7038865141000192, 1699612131395493888, -3459036721655810048, -1025681798088053424128

Offset: 0

Seiichi Manyama, May 03 2024

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			B(x) = x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 + 256*x^7 + 768*x^8 - 14848*x^9 + 51200*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..435
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.

Cf. A002697, A309509.

Let B(x) = A(A(x)). B(B(x)) = F(x).

B(x) = G(2*x)/2, where G(x) is the g.f. for A309509.

A372499 G.f. satisfies A(A(A(x))) = F(x), where F(x) is the g.f. for A053540(n) = n*9^(n-1).

Original entry on oeis.org

0, 1, 6, 9, 54, 0, -1944, 44469, -323676, -5990193, 179194032, 484654509, -105337511100, 757846026261, 85419734244300, -1707846638480514, -90276038133498612, 3464956887464464164, 118426852966952180502, -7984363576091338944720, -181143285020960488524558

Offset: 0

Seiichi Manyama, May 03 2024

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			A(A(x)) = x + 12*x^2 + 90*x^3 + 594*x^4 + 3807*x^5 + 20412*x^6 + 123201*x^7 + 1032264*x^8 - 1463103*x^9 - 35468766*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A309509, A372492.

Cf. A053540, A141118, A372500.

Define the sequence b(n,m) as follows. If n

A372522 G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = Sum_{k>=1} k * 18^(k-1) * x^k.

Original entry on oeis.org

0, 1, 6, -18, 378, -5670, 52488, 930204, -55108026, 575622774, 46483766460, -1494416264796, -85327731650772, 5947844644410876, 192190798316367540, -29067440301002581416, -418574641900663175706, 179341053539746099078422

Offset: 0

Seiichi Manyama, May 04 2024

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			A(A(x)) = x + 12*x^2 + 36*x^3 + 432*x^4 - 62208*x^6 + 2846016*x^7 - ...
A(A(A(x))) = x + 18*x^2 + 162*x^3 + 1458*x^4 + 13122*x^5 + 2125764*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A309509, A372492, A372499, A372520.

Define the sequence b(n,m) as follows. If n

Let F(x) = x/(1 - 18*x)^2, B(x) = A(A(x)) and C(x) = A(A(A(x))).

B(B(B(x))) = C(C(x)) = F(x).

B(x) = G(2*x)/2, where G(x) is the g.f. for A372499.

C(x) = H(9*x)/9, where H(x) is the g.f. for A309509.

A372493 G.f. A(x) satisfies A(A(x)) = Sum_{k>=1} k^2 * 2^(k-1) * x^k.

Original entry on oeis.org

0, 1, 4, 2, 12, -30, 24, 1412, -18716, 127750, -19448, -10721556, 98983992, 546122580, -19718580272, 30721006440, 4638904011364, -44204880124922, -1218793973236472, 26364933421291468, 327900715232299304, -15425392878552410820, -62258050574118828336

Offset: 0

Seiichi Manyama, May 03 2024

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Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986 [math.CO], 2013.

Cf. A014477, A309509.

A372520 G.f. A(x) satisfies A(A(A(A(A(x))))) = Sum_{k>=1} k * 25^(k-1) * x^k.

Original entry on oeis.org

0, 1, 10, -25, 1000, -18125, 131250, 11609375, -630156250, 4314062500, 1173535156250, -38006699218750, -4262573730468750, 321379049072265625, 20787043081054687500, -3209395283374023437500, -116229452332824707031250, 39638105812041778564453125

Offset: 0

Seiichi Manyama, May 04 2024

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Cf. A309509, A372492, A372499, A372522.

Cf. A141120.

Define the sequence b(n,m) as follows. If n

Showing 1-5 of 5 results.

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A372492 G.f. satisfies A(A(A(A(x)))) = F(x), where F(x) is the g.f. for A002697(n) = n*4^(n-1).

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A372499 G.f. satisfies A(A(A(x))) = F(x), where F(x) is the g.f. for A053540(n) = n*9^(n-1).

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A372522 G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = Sum_{k>=1} k * 18^(k-1) * x^k.

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A372493 G.f. A(x) satisfies A(A(x)) = Sum_{k>=1} k^2 * 2^(k-1) * x^k.

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A372520 G.f. A(x) satisfies A(A(A(A(A(x))))) = Sum_{k>=1} k * 25^(k-1) * x^k.

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