A000621 -id:A000621 - cp's OEIS frontend

A352006 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 2^k * a(k) * a(n-2*k-1).

1, 1, 1, 3, 5, 11, 21, 59, 117, 283, 597, 1467, 3125, 7387, 16149, 39931, 87541, 207643, 463061, 1107515, 2473909, 5819739, 13132437, 31080571, 70236533, 164315035, 373572693, 875121339, 1991869237, 4639482331, 10599986709, 24765957371, 56617082101

Offset: 0

Ilya Gutkovskiy, Feb 28 2022

nonn

Cf. A000621, A015083, A352007, A352008.

a[0] = 1; a[n_] := a[n] = Sum[2^k a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 32}]
nmax = 32; A[] = 0; Do[A[x] = 1/(1 - x A[2 x^2]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(2*x^2)).

A352007 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 3^k * a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 1, 1, 4, 7, 19, 40, 178, 379, 1237, 2941, 10378, 24628, 78928, 198820, 813550, 1971907, 6587245, 16980079, 61488286, 155573011, 515316037, 1363261084, 4937498686, 12796438252, 42078038668, 113153315824, 390012381346, 1036020692356, 3379994401042, 9240830253940

Offset: 0

Ilya Gutkovskiy, Feb 28 2022

nonn

Cf. A000621, A015084, A352006, A352008.

a[0] = 1; a[n_] := a[n] = Sum[3^k a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 30}]
nmax = 30; A[] = 0; Do[A[x] = 1/(1 - x A[3 x^2]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(3*x^2)).

A352008 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 1, 1, 5, 9, 29, 65, 437, 953, 3981, 10097, 49829, 123241, 516349, 1400737, 10203285, 24698905, 111642477, 304787665, 1704790917, 4392726473, 19951366877, 56296655617, 336083829621, 878995865721, 3974885167949, 11362790432305, 60789762148453, 165051865924137

Offset: 0

Ilya Gutkovskiy, Feb 28 2022

nonn

Cf. A000621, A015085, A352006, A352007.

a[0] = 1; a[n_] := a[n] = Sum[4^k a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 28}]
nmax = 28; A[] = 0; Do[A[x] = 1/(1 - x A[4 x^2]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(4*x^2)).

A036677 Used by Polya in calculating A000598.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 5, 19, 63, 184, 503, 1300, 3229, 7748, 18106, 41327, 92521, 203601, 441466, 944618, 1997726, 4180405, 8665024, 17805427, 36299704, 73468100, 147704332, 295122569, 586304222, 1158582080, 2278084748, 4458522727

Offset: 0

N. J. A. Sloane

nonn

G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; q^{II}(x) on p. 441.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; q^{II}(x) on p. 441. (Annotated scanned copy)

G.f.: x * q(x) * (2*q(x)*p(x^2) + (q(x)^2-q(x^2)) * p(x)) / 2 where p(x) is the g.f. for A036676 and q(x) is the g.f. for A000621 with offset 0 [from Polya]. - Sean A. Irvine, Nov 21 2020

A217925 G.f. A(x) satisfies A(x) = 1 + xA(x)A(x^2)^2.

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 19, 40, 77, 155, 306, 610, 1207, 2400, 4760, 9456, 18765, 37257, 73955, 146813, 291434, 578524, 1148434, 2279720, 4525487, 8983421, 17832976, 35399824, 70271944, 139495472, 276910976, 549691232, 1091185133, 2166094309, 4299884233, 8535634803, 16943967775

Offset: 0

Joerg Arndt, Oct 15 2012

nonn

What does this sequence count?

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000108 (A(x) = 1 + x*A(x)^2), A000621 (A(x) = 1 + x*A(x)*A(x^2)).

Cf. A036675 (A(x) = 1 + x*A(x)^2*A(x^2)), A101913 (A(x) = 1 + x*A(x)*A(x^3); for abs. values).

T(n,m):=if n=m then 1 else sum(binomial(m+k-1,k)*T((n-m)/2,2*k),k,1,(n-m)/4);
makelist(T(4*n+1,1),n,0,25); /* Vladimir Kruchinin, Mar 25 2015 */

N=66;  R=O('x^N);  x='x+R;
F = 1 + x;
{ for (k=1,N+1, F = 1 + x * F * subst(F,'x,'x^2)^2 + R; ); }
Vec(F+O('x^N))

a(n) ~ c * d^n, where d = 1.985085392419660786124534041173530134614822710253953085885966352..., c = 0.322822740100478716884116064042886830242825005622702339543369128... . - Vaclav Kotesovec, Aug 10 2014

a(n) = T(4*n+1,1), where T(n,m) = Sum_{k=1..(n-m)/4} C(m+k-1,k)*T((n-m)/2,2*k). - Vladimir Kruchinin, Mar 25 2015

A036678 Used by Polya in calculating A000598.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 7, 37, 154, 558, 1830, 5581, 16077, 44251, 117308, 301330, 753458, 1840568, 4405344, 10355493, 23953832, 54614151, 122903818, 273320540, 601273377, 1309641376, 2826528077, 6048868456, 12843409063, 27071247989

Offset: 0

N. J. A. Sloane

nonn

G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; q^{III}(x) on p. 441.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; q^{III}(x) on p. 441. (Annotated scanned copy)

G.f.: x * q(x) * (2*p(x)*p(x^2) + (q(x)^2-q(x^2))*m(x) + (p(x)^2-p(x^2))*q(x)) / 2 where m(x) is the g.f. of A036677, p(x) is the g.f. of A036676, and q(x) is the g.f. of A000621 with offset 0 [from Polya]. - Sean A. Irvine, Nov 21 2020

A351938 a(0) = 1; a(n) = -a(n-1) + 2 * Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 17, 35, 65, 125, 241, 463, 885, 1701, 3265, 6271, 12033, 23105, 44353, 85147, 163445, 313777, 602353, 1156339, 2219809, 4261389, 8180561, 15704215, 30147333, 57873821, 111100225, 213278943, 409431169, 785984353, 1508852673, 2896541859

Offset: 0

Ilya Gutkovskiy, Feb 26 2022

nonn

Cf. A000621, A001003, A319436, A349365.

a[0] = 1; a[n_] := a[n] = -a[n - 1] + 2 Sum[a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 35}]
nmax = 35; A[] = 0; Do[A[x] = 1/(1 + x - 2 x A[x^2]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(x^2)).

cp's OEIS Frontend

A352006 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 2^k * a(k) * a(n-2*k-1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A352007 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 3^k * a(k) * a(n-2*k-1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A352008 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * a(k) * a(n-2*k-1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A036677 Used by Polya in calculating A000598.

Views

Author

Keywords

Links

Formula

A217925 G.f. A(x) satisfies A(x) = 1 + x*A(x)*A(x^2)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maxima

PARI

Formula

A036678 Used by Polya in calculating A000598.

Views

Author

Keywords

Links

Formula

A351938 a(0) = 1; a(n) = -a(n-1) + 2 * Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A217925 G.f. A(x) satisfies A(x) = 1 + xA(x)A(x^2)^2.