A115055 -id:A115055 - cp's OEIS frontend

A291382 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2.

2, 7, 22, 70, 222, 705, 2238, 7105, 22556, 71608, 227332, 721705, 2291178, 7273743, 23091762, 73308814, 232731578, 738846865, 2345597854, 7446508273, 23640235416, 75050038224, 238259397096, 756395887969, 2401310279090, 7623377054503, 24201736119310

Offset: 0

Clark Kimberling, Sep 04 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
In the following guide to p-INVERT sequences using s = (1,1,0,0,0,...) = A019590, in some cases t(1,1,0,0,0,...) is a shifted version of the cited sequence:
p(S)                     t(1,1,0,0,0,...)
1 - S                       A000045 (Fibonacci numbers)
1 - S^2                     A094686
1 - S^3                     A115055
1 - S^4                     A291379
1 - S^5                     A281380
1 - S^6                     A281381
1 - 2 S                     A002605
1 - 3 S                     A125145
(1 - S)^2                   A001629
(1 - S)^3                   A001628
(1 - S)^4                   A001629
(1 - S)^5                   A001873
(1 - S)^6                   A001874
1 - S - S^2                 A123392
1 - 2 S - S^2               A291382
1 - S - 2 S^2               A124861
1 - 2 S - S^2               A291383
(1 - 2 S)^2                 A073388
(1 - 3 S)^2                 A291387
(1 - 5 S)^2                 A291389
(1 - 6 S)^2                 A291391
(1 - S)(1 - 2 S)            A291393
(1 - S)(1 - 3 S)            A291394
(1 - 2 S)(1 - 3 S)          A291395
(1 - S)(1 - 2 S)            A291393
(1 - S)(1 - 2 S)(1 - 3 S)   A291396
1 - S - S^3                 A291397
1 - S^2 - S^3               A291398
1 - S - S^2 - S^3           A186812
1 - S - S^2 - S^3 - S^4     A291399
1 - S^2 - S^4               A291400
1 - S - S^4                 A291401
1 - S^3 - S^4               A291402
1 - 2 S^2 - S^4             A291403
1 - S^2 - 2 S^4             A291404
1 - 2 S^2 - 2 S^4           A291405
1 - S^3 - S^6               A291407
(1 - S)(1 - S^2)            A291408
(1 - S^2)(1 - S)^2          A291409
1 - S - S^2 - 2 S^3         A291410
1 - 2 S - S^2 + S^3         A291411
1 - S - 2 S^2 + S^3         A291412
1 - 3 S + S^2 + S^3         A291413
1 - 2 S + S^3               A291414
1 - 3 S + S^2               A291415
1 - 4 S + S^2               A291416
1 - 4 S + 2 S^2             A291417

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 3, 2, 1)

Cf. A019590, A290890, A291000, A291219, A291728, A292479, A292480.

z = 60; s = x + x^2; p = 1 - 2 s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291382 *)

G.f.: (-2 - 3 x - 2 x^2 - x^3)/(-1 + 2 x + 3 x^2 + 2 x^3 + x^4).

a(n) = 2*a(n-1) + 3*a(n-2) + 2*a(n-3) + a(n-4) for n >= 5.

A361568 Expansion of e.g.f. exp(x^3/6 * (1+x)^3).

Original entry on oeis.org

1, 0, 0, 1, 12, 60, 130, 420, 8400, 101080, 781200, 4435200, 37714600, 607807200, 8660652000, 94007313400, 914497584000, 11566931376000, 198256136478400, 3275456501116800, 46558791351072000, 636647461257808000, 10238792220969312000, 194852563745775936000

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A047974, A361567, A361569.

Cf. A115055, A361279.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/6*(1+x)^3)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=3, i, j*binomial(3, j-3)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k)/(6^k * k!).
a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} k * binomial(3,k-3) * a(n-k)/(n-k)!.
a(n) = (n-1)*(n-2)/6 * (3*a(n-3) + 12*(n-3)*a(n-4) + 15*(n-3)*(n-4)*a(n-5) + 6*(n-3)*(n-4)*(n-5)*a(n-6)). -Seiichi Manyama, Jun 16 2024

A361730 Diagonal of rational function 1/(1 - (1 + xyz) * (x^3 + y^3 + z^3)).

Original entry on oeis.org

1, 0, 0, 6, 18, 18, 96, 540, 1350, 3480, 16470, 61020, 175860, 627480, 2498580, 8520876, 28563570, 106917300, 393495396, 1369171188, 4914119826, 18191218716, 65741140080, 235643531508, 862450963704, 3163777886412, 11484836808588, 41875694151720

Offset: 0

Seiichi Manyama, Mar 22 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..1723

Cf. A361728, A361729.

Cf. A115055.

a(n) = sum(k=0, n\3, (3*k)!/k!^3*binomial(3*k, n-3*k));

a(n) = Sum_{k=0..floor(n/3)} (3*k)!/k!^3 * binomial(3*k,n-3*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: (n-1)*n^2*a(n) = -(n-1)^2*n*a(n-1) + 27*(n-2)*(n-1)^2*a(n-3) + 108*(n-2)*(n^2 - 3*n + 1)*a(n-4) + 54*(3*n^3 - 18*n^2 + 28*n - 5)*a(n-5) + 108*(n^3 - 7*n^2 + 12*n - 1)*a(n-6) + 27*(n-5)*(n-3)*n*a(n-7).
a(n) ~ sqrt(3) * ((3 + sqrt(21))/2)^n / (2*Pi*n). (End)

A378151 G.f. A(x) satisfies A(x) = 1 + (x * (1+x) * A(x))^3.

Original entry on oeis.org

1, 0, 0, 1, 3, 3, 4, 18, 45, 72, 153, 450, 1066, 2172, 5142, 13381, 31752, 72333, 176475, 441909, 1065528, 2551465, 6292857, 15620439, 38229235, 93698523, 232545105, 578019090, 1430290512, 3548336724, 8851036863, 22092054588, 55093739760, 137681640450

Offset: 0

Seiichi Manyama, Nov 18 2024

nonn

Cf. A000045, A256169, A378152.

Cf. A115055, A378153.

a(n) = sum(k=0, n\3, binomial(3*k, n-3*k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k) * binomial(3*k,k)/(2*k+1).

A375318 Expansion of 1/(1 - x^3*(1 + x)^4).

Original entry on oeis.org

1, 0, 0, 1, 4, 6, 5, 9, 28, 57, 82, 122, 249, 519, 913, 1485, 2632, 5053, 9369, 16375, 28662, 52226, 96182, 173220, 307653, 551927, 1002327, 1815191, 3258813, 5845015, 10539893, 19048900, 34332648, 61735922, 111129005, 200406479, 361364501, 650804074, 1171717523

Offset: 0

Seiichi Manyama, Aug 12 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,1,4,6,4,1).

Cf. A115055, A264622.

Cf. A375314.

my(N=40, x='x+O('x^N)); Vec(1/(1-x^3*(1+x)^4))

a(n) = sum(k=0, n\3, binomial(4*k, n-3*k));

a(n) = a(n-3) + 4*a(n-4) + 6*a(n-5) + 4*a(n-6) + a(n-7).

a(n) = Sum_{k=0..floor(n/3)} binomial(4*k,n-3*k).

A375319 Expansion of (1 + x)/(1 - x^3*(1 + x)^3).

Original entry on oeis.org

1, 1, 0, 1, 4, 6, 5, 8, 21, 36, 45, 66, 128, 224, 330, 497, 851, 1452, 2287, 3556, 5826, 9693, 15624, 24807, 40126, 65737, 106584, 171112, 276160, 448980, 728201, 1174985, 1897380, 3074733, 4982688, 8055918, 13020029, 21074012, 34125561, 55210580, 89284541

Offset: 0

Seiichi Manyama, Aug 12 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,1,3,3,1).

Cf. A115055, A375321.

my(N=50, x='x+O('x^N)); Vec((1+x)/(1-x^3*(1+x)^3))

a(n) = sum(k=0, n\3, binomial(3*k+1, n-3*k));

a(n) = a(n-3) + 3*a(n-4) + 3*a(n-5) + a(n-6).

a(n) = Sum_{k=0..floor(n/3)} binomial(3*k+1,n-3*k).

A375321 Expansion of (1 + x)^2/(1 - x^3*(1 + x)^3).

Original entry on oeis.org

1, 2, 1, 1, 5, 10, 11, 13, 29, 57, 81, 111, 194, 352, 554, 827, 1348, 2303, 3739, 5843, 9382, 15519, 25317, 40431, 64933, 105863, 172321, 277696, 447272, 725140, 1177181, 1903186, 3072365, 4972113, 8057421, 13038606, 21075947, 34094041, 55199573, 89336141

Offset: 0

Seiichi Manyama, Aug 12 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,1,3,3,1).

Cf. A115055, A375319.

Cf. A375317,

my(N=40, x='x+O('x^N)); Vec((1+x)^2/(1-x^3*(1+x)^3))

a(n) = sum(k=0, n\3, binomial(3*k+2, n-3*k));

a(n) = a(n-3) + 3*a(n-4) + 3*a(n-5) + a(n-6).
a(n) = Sum_{k=0..floor(n/3)} binomial(3*k+2,n-3*k).
a(n) = A375319(n) + A375319(n-1).

A378153 G.f. A(x) satisfies A(x) = 1 + (x * (1+x))^3 * A(x)^2.

Original entry on oeis.org

1, 0, 0, 1, 3, 3, 3, 12, 30, 45, 75, 192, 436, 798, 1554, 3542, 7740, 15543, 32183, 70794, 153252, 321431, 684123, 1491504, 3232672, 6928779, 14957787, 32615388, 70991040, 153985890, 335256886, 733206840, 1603258134, 3503385568, 7671749664, 16837946850

Offset: 0

Seiichi Manyama, Nov 18 2024

nonn

Cf. A115055, A378151.

Cf. A000108.

a(n) = sum(k=0, n\3, binomial(3*k, n-3*k)*binomial(2*k, k)/(k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k) * C(k), where C(k) are the Catalan numbers (A000108).

G.f.: 2/(1 + sqrt(1 - 4*(x*(1+x))^3)).

cp's OEIS Frontend

A291382 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2.

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A361568 Expansion of e.g.f. exp(x^3/6 * (1+x)^3).

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A361730 Diagonal of rational function 1/(1 - (1 + x*y*z) * (x^3 + y^3 + z^3)).

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A378151 G.f. A(x) satisfies A(x) = 1 + (x * (1+x) * A(x))^3.

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A375318 Expansion of 1/(1 - x^3*(1 + x)^4).

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A375319 Expansion of (1 + x)/(1 - x^3*(1 + x)^3).

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PARI

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A375321 Expansion of (1 + x)^2/(1 - x^3*(1 + x)^3).

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A378153 G.f. A(x) satisfies A(x) = 1 + (x * (1+x))^3 * A(x)^2.

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A361730 Diagonal of rational function 1/(1 - (1 + xyz) * (x^3 + y^3 + z^3)).