A171200 -id:A171200 - cp's OEIS frontend

A135867 G.f. satisfies A(x) = 1 + xA(2x)^2.

1, 1, 4, 36, 640, 21888, 1451008, 188941312, 48768745472, 25069815595008, 25722272102744064, 52730972085034156032, 216091838647321476726784, 1770657164881170759078117376, 29013990909330956353981535748096

Offset: 0

Paul D. Hanna, Dec 02 2007

nonn

Self-convolution equals A135868 such that 2^n*A135868(n) = a(n+1) for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..81

Cf. A015083, A135868, A171192.

Cf. A171200, A171202, A171204, A171206, A171208, A171210, A343439.

nmax = 15; A[] = 0; Do[A[x] = 1 + x*A[2*x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 04 2021 *)

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,2*x)^2);polcoeff(A,n)}

a(n)=if(n==0,1,2^(n-1)*sum(k=0,n-1,a(k)*a(n-k-1))) \\ Paul D. Hanna, Feb 09 2010

a(n) = 2^(n-1)*Sum_{k=0..n-1} a(k)*a(n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Feb 09 2010

a(n) ~ c * 2^(n*(n+1)/2), where c = 0.715337433614869740944075474484711589980951273610257702786245519231799678... - Vaclav Kotesovec, Nov 04 2021

A171211 G.f. satisfies: A(x) = (1 + x*A(2x))^8.

Original entry on oeis.org

1, 8, 156, 5944, 425286, 57811000, 15246040860, 7918843018760, 8165859019876353, 16781575370067304448, 68855523432488884833408, 564547878300963670909315840, 9253510119618208634494942344960

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Cf. A135868, A171200-A171209, A171210.

m = 13; A[] = 0; Do[A[x] = (1 + x A[2 x])^8 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=(1+x*subst(A, x, 2*x))^8); polcoeff(A, n)}

Self-convolution 8th power of A171210 where a(n) = A171210(n+1)/2^n for n>=0.

A143046 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^3.

Original entry on oeis.org

1, 1, -3, -6, 35, 87, -588, -1578, 11511, 32223, -245883, -706824, 5556564, 16267508, -130617600, -387533058, 3161190783, 9474886287, -78241316361, -236394953670, 1971270824859, 5994591989967, -50388913722480, -154052058035736

Offset: 0

Paul D. Hanna, Jul 19 2008

sign

			G.f.: A(x) = 1 + x - 3*x^2 - 6*x^3 + 35*x^4 + 87*x^5 - 588*x^6 - 1578*x^7 +...
where
A(x)^3 = 1 + 3*x - 6*x^2 - 35*x^3 + 87*x^4 + 588*x^5 - 1578*x^6 - 11511*x^7 +...
A(x)^4 = 1 + 4*x - 6*x^2 - 56*x^3 + 87*x^4 + 1008*x^5 - 1578*x^6 - 20464*x^7 +...
Note that a bisection of A^4 equals a bisection of A^3.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A143045, A143047, A143048, A143049, A213252, A213281, A213335.

Cf. A171200.

a(n)=local(A=x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,-x)^3);polcoeff(A,n)

G.f. satisfies: A(x) = 1 + x*(1 - x*A(x)^3)^3.
G.f. satisfies: [A(x)^4 + A(-x)^4]/2 = [A(x)^3 + A(-x)^3]/2.
a(0) = 1; a(n) = (-1)^(n-1) * Sum_{i, j, k>=0 and i+j+k=n-1} a(i) * a(j) * a(k). - Seiichi Manyama, Jul 08 2025

A171202 G.f. A(x) satisfies A(x) = 1 + xA(2x)^4.

Original entry on oeis.org

1, 1, 8, 152, 5664, 399376, 53846016, 14141384704, 7330134466560, 7551251740344320, 15510852680588984320, 63626087316632048238592, 521607805205244557347782656, 8549156556447111748331767857152, 280190094729160875643888549840814080, 18364219805837823940403573170370661842944

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A135867, A171200, A171201, A171203, A171204-A171211, A343439.

Cf. A143047, A171193.

terms = 16; A[] = 0; Do[A[x] = 1 + x*A[2x]^4 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^4); polcoeff(A, n)}

a(0) = 1; a(n) = 2^(n-1) * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 08 2025

A171204 G.f. A(x) satisfies A(x) = 1 + xA(2x)^5.

Original entry on oeis.org

1, 1, 10, 240, 11280, 1000080, 169100832, 55605632640, 36058105605120, 46450803286978560, 119290436529298554880, 611727201854914747760640, 6268994998754867059071385600, 128439243721180540266999017635840, 5261899692949082390205726962630000640, 431096933496167311430326245852780460769280

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..79

Cf. A135867, A171200-A171203, A171205, A171206-A171211, A343439.

Cf. A143048, A171194.

terms = 16; A[] = 0; Do[A[x] = 1 + x*A[2x]^5 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^5); polcoeff(A, n)}

a(0) = 1; a(n) = 2^(n-1) * Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 08 2025

A171206 G.f. A(x) satisfies A(x) = 1 + xA(2x)^6.

Original entry on oeis.org

1, 1, 12, 348, 19744, 2108784, 428817600, 169398274624, 131889504749568, 203937600707475456, 628561895904796999680, 3868208404121906515820544, 47571342639450113377565933568, 1169589733863427138021074362433536, 57499379103783344787572704263568097280, 5652994168279651703590653986228287051923456

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..79

Cf. A135867, A171200-A171205, A171207, A171208-A171211, A343439.

Cf. A143049, A171195.

terms = 16; A[] = 0; Do[A[x] = 1+x*A[2x]^6 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^6); polcoeff(A, n)}

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 08 2025

A171208 G.f. A(x) satisfies A(x) = 1 + xA(2x)^7.

Original entry on oeis.org

1, 1, 14, 476, 31640, 3953488, 939383200, 433281169216, 393718899904640, 710399428248892928, 2554705943898166145024, 18342976469146094416494592, 263185684727811758287894478848, 7549222852919288301041224694890496, 432993292623369448352459156263293419520

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..79

Cf. A135867, A171200-A171207, A171209, A171210-A171211, A343439.

Cf. A171196.

terms = 15; A[] = 0; Do[A[x] = 1+x*A[2x]^7 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^7); polcoeff(A, n)}

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 08 2025

A171210 G.f. A(x) satisfies A(x) = 1 + xA(2x)^8.

Original entry on oeis.org

1, 1, 16, 624, 47552, 6804576, 1849952000, 975746615040, 1013611906401280, 2090459909088346368, 8592166589474459877376, 70508055994868618069409792, 1156194054760373598022278840320, 37902377449956182566891283844956160, 2484501232375923934830943089632156319744

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..79

Cf. A135867, A171200-A171209, A171211, A343439.

Cf. A171197.

terms = 15; A[] = 0; Do[A[x] = 1+x*A[2x]^8 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^8); polcoeff(A, n)}

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_8>=0 and x_1+x_2+...+x_8=n-1} Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 08 2025

A171201 G.f. satisfies: A(x) = (1 + x*A(2x))^3.

Original entry on oeis.org

1, 3, 21, 289, 7566, 380424, 37361616, 7252471584, 2799853666176, 2155959119115264, 3315891500224031232, 10193070293871040606464, 62646640175842537242599936, 769927299959295414569740867584, 18923273743619678311418282019397632, 930154604531789703005691292148132511744

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Cf. A135868, A171200, A171202-A171211.

terms = 16; A[] = 0; Do[A[x] = (1 + x*A[2x])^3 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=(1+x*subst(A, x, 2*x))^3); polcoeff(A, n)}

Self-convolution cube of A171200 where a(n) = A171200(n+1)/2^n for n>=0.

a(14)-a(15) from Stefano Spezia, Apr 02 2025

A171203 G.f. satisfies: A(x) = (1 + x*A(2x))^4.

Original entry on oeis.org

1, 4, 38, 708, 24961, 1682688, 220959136, 57266675520, 29497077110720, 30294634141775360, 62134850895148484608, 254691311135373319017472, 2087196424913845641682560512, 34202892422993270952623113994240, 1120863025258656246362522776511881216, 73460242428855296330451249854756580540416

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Cf. A135868, A171200, A171201, A171202, A171204-A171211.

terms = 16; A[] = 0; Do[A[x] = (1 + x*A[2x])^4 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=(1+x*subst(A, x, 2*x))^4); polcoeff(A, n)}

Self-convolution 4th power of A171202 where a(n) = A171202(n+1)/2^n for n>=0.

a(14)-a(15) from Stefano Spezia, Apr 02 2025

cp's OEIS Frontend

A135867 G.f. satisfies A(x) = 1 + x*A(2*x)^2.

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A171211 G.f. satisfies: A(x) = (1 + x*A(2x))^8.

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A143046 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^3.

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A171202 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^4.

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A171204 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^5.

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A171206 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^6.

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A171208 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^7.

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A171210 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^8.

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A135867 G.f. satisfies A(x) = 1 + xA(2x)^2.

A171202 G.f. A(x) satisfies A(x) = 1 + xA(2x)^4.

A171204 G.f. A(x) satisfies A(x) = 1 + xA(2x)^5.

A171206 G.f. A(x) satisfies A(x) = 1 + xA(2x)^6.

A171208 G.f. A(x) satisfies A(x) = 1 + xA(2x)^7.

A171210 G.f. A(x) satisfies A(x) = 1 + xA(2x)^8.