cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A145350 G.f. satisfies: A(x/A(x)^2) = 1 + x*A(x).

Original entry on oeis.org

1, 1, 3, 18, 154, 1632, 20007, 273164, 4058556, 64628487, 1091488334, 19403175105, 361028420037, 7000932594042, 141010975529568, 2942134448306481, 63449975020918843, 1411787024678728344, 32360032648643379471, 763096191377494726161, 18491954778730596443088
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2008

Keywords

Comments

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 154*x^4 + 1632*x^5 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 42*x^3 + 353*x^4 + 3680*x^5 + 44526*x^6+...
A(x/A(x)^2) = 1 + x + x^2 + 3*x^3 + 18*x^4 + 154*x^5 + 1632*x^6 +...
A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)^2):
G(x) = 1 + x + 5*x^2 + 41*x^3 + 432*x^4 + 5329*x^5 + 73512*x^6 +...
G(x)^2 = 1 + 2*x + 11*x^2 + 92*x^3 + 971*x^4 + 11932*x^5 +...
To illustrate the formula a(n) = [x^(n-1)] 3*A(x)^(2*n+1)/(2*n+1),
form a table of coefficients in A(x)^(2*n+1) as follows:
A^3: [(1), 3, 12, 73, 606, 6225, 74370, 994668, ...];
A^5: [1, (5), 25, 160, 1315, 13191, 153930, 2017620, ...];
A^7: [1, 7, (42), 287, 2373, 23436, 267988, 3445835, ...];
A^9: [1, 9, 63, (462), 3888, 38106, 428637, 5414760, ...];
A^11: [1, 11, 88, 693, (5984), 58619, 651354, 8099410, ...];
A^13: [1, 13, 117, 988, 8801, (86697), 955656, 11723712, ...];
A^15: [1, 15, 150, 1355, 12495, 124398, (1365820), 16571385, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[3/3*(1), 3/5*(5), 3/7*(42), 3/9*(462), 3/11*(5984), 3/13*(86697), ...].
ALTERNATE GENERATING METHOD.
This sequence forms column zero in the follow array.
Let A denote this sequence, and A^2 the self-convolution square of A.
Start in row zero with this sequence, A, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A^2 and the remaining terms in a given row to obtain the next row:
[1, 1, 1, 3, 18, 154, 1632, 20007, 273164, 4058556, 64628487, ...];
[1, 3, 12, 73, 606, 6225, 74370, 994668, 14535285, 228349287, ...];
[3, 18, 118, 962, 9511, 109404, 1415942, 20128565, 309001962, ...];
[18, 154, 1324, 13017, 146470, 1849625, 25701033, 386747469, ...];
[154, 1632, 16743, 188240, 2343654, 32006379, 473572975, ...];
[1632, 20007, 233150, 2905879, 39290669, 573813430, 8978918475, ...];
[20007, 273164, 3512228, 47574771, 689590692, 10679554646, ...];
[273164, 4058556, 56511375, 820798718, 12635699895, ...];
[4058556, 64628487, 962231360, 14843336308, 241004566025, ...]; ...

Links

Crossrefs

Cf. variants: A001764, A147664, A120972, A145345, A145349, A147664.

Programs

  • PARI
    {a(n)=local(A=1+x,G);for(i=0,n,G=(serreverse(x/(A+x*O(x^n))^2)/x)^(1/2);A=1+x*G^3);polcoeff(A,n)}
  • PARI
    /* This sequence is generated when k=2, m=1: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=2, m=1)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* Prints terms 0..30 */
{A=[1];
for(m=1,30,
  B=A;
  for(i=1,m-1, C=Vec(Ser(A)^2*Ser(B)); B=vector(#C-1,n,C[n+1]) );
  A=concat(A,0);A[#A]=B[1]
);
A} \\ Paul D. Hanna, Jan 10 2016

Formula

G.f.: A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)^2) and A(x) = G(x/A(x)^2).
a(n) = [x^(n-1)] 3*A(x)^(2*n+1)/(2*n+1) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 3*A(x)^(2*n+1)/(2*n+1) for n>=1.

A147664 G.f. satisfies: A(x/A(x)) = 1 + x*A(x)^2.

Original entry on oeis.org

1, 1, 3, 15, 100, 801, 7296, 73174, 791751, 9116613, 110640310, 1405349658, 18585016509, 254855278565, 3612425924919, 52793266545585, 793851646358364, 12261570084250926, 194260753173421656, 3153098224666860712
Offset: 0

Views

Author

Paul D. Hanna, Nov 09 2008

Keywords

Comments

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 100*x^4 + 801*x^5 + 7296*x^6 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 36*x^3 + 239*x^4 + 1892*x^5 + 17019*x^6 +...
A(x/A(x)) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 239*x^5 + 1892*x^6 +...
A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)) is the g.f. of A182953:
G(x) = 1 + x + 4*x^2 + 25*x^3 + 197*x^4 + 1797*x^5 + 18178*x^6 +...
To illustrate the formula a(n) = [x^(n-1)] 3*A(x)^(n+2)/(n+2),
form a table of coefficients in A(x)^(n+2) as follows:
A^3: [(1), 3, 12, 64, 426, 3345, 29766, 291999, ...];
A^4: [1, (4), 18, 100, 671, 5244, 46248, 449264, ...];
A^5: [1, 5, (25), 145, 985, 7686, 67305, 648085, ...];
A^6: [1, 6, 33, (200), 1380, 10782, 93922, 897402, ...];
A^7: [1, 7, 42, 266, (1869), 14658, 127246, 1207753, ...];
A^8: [1, 8, 52, 344, 2466, (19456), 168604, 1591496, ...];
A^9: [1, 9, 63, 435, 3186, 25335, (219522), 2063052, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[3/3*(1), 3/4*(4), 3/5*(25), 3/6*(200), 3/7*(1869), 3/8*(19456), ...].
ALTERNATE GENERATING METHOD.
This sequence forms column zero in the follow array.
Let A denote this sequence, and A^2 the self-convolution square of A.
Start in row zero with A^2, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A and the remaining terms in a given row to obtain the next row:
[1, 1, 2, 7, 36, 239, 1892, 17019, 168746, 1807656, 20634852, 248560373, ...];
[1, 3, 12, 64, 426, 3345, 29766, 291999, 3097746, 35059659, 419160576,...];
[3, 15, 85, 571, 4443, 38952, 376090, 3930156, 43875903, 518191486, ...];
[15, 100, 701, 5494, 47883, 457451, 4724372, 52138050, 609215321, ...];
[100, 801, 6495, 56980, 542331, 5558082, 60793521, 704009142, ...];
[801, 7296, 65878, 630811, 6448878, 70124397, 806356465, 9749112990, ...];
[7296, 73174, 718577, 7386763, 80183478, 917721557, 11031656810, ...];
[73174, 791751, 8324862, 90823582, 1038161379, 12431200320, 155525155360, ...]; ...

Links

Crossrefs

Cf. A182953, A145348, A145350, A120972, A145345, A145349.

Programs

  • PARI
    {a(n)=local(F=1+x,G);for(i=0,n,G=serreverse(x/(F+x*O(x^n))^1)/x;F=1+x*G^3);polcoeff(F,n)}
  • PARI
    /* This sequence is generated when k=1, m=2: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=1, m=2)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* Prints terms 0..30 */
{A=[1];
for(m=1,30,
  B=Vec(Ser(A)^2);
  for(i=1,m-1, C=Vec(Ser(A)*Ser(B)); B=vector(#C-1,n,C[n+1]) );
  A=concat(A,0); A[#A]=B[1];
);
A} \\ Paul D. Hanna, Jan 10 2016

Formula

G.f.: A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)) and A(x) = G(x/A(x)) is the g.f. of A182953.
a(n) = [x^(n-1)] 3*A(x)^(n+2)/(n+2) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 3*A(x)^(n+2)/(n+2) for n>=1 (see comment).

A145349 G.f. satisfies: A(x/A(x)^3) = 1 + x*A(x).

Original entry on oeis.org

1, 1, 4, 34, 416, 6319, 111124, 2177346, 46440184, 1061938195, 25762345804, 658072997702, 17600573291712, 490770914734054, 14219015899154068, 426904437068035200, 13252855203929697200, 424634035832800883743, 14020984603588221319072, 476460483567807053745450
Offset: 0

Views

Author

Paul D. Hanna, Nov 11 2008

Keywords

Comments

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 416*x^4 + 6319*x^5 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 127*x^3 + 1512*x^4 + 22419*x^5 +...
A(x/A(x)^3) = 1 + x + x^2 + 4*x^3 + 34*x^4 + 416*x^5 + 6319*x^6 +...
A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^3):
G(x) = 1 + x + 7*x^2 + 82*x^3 + 1239*x^4 + 21942*x^5 + 434746*x^6 +...
G(x)^3 = 1 + 3*x + 24*x^2 + 289*x^3 + 4377*x^4 + 77097*x^5 +...
To illustrate the formula a(n) = [x^(n-1)] 4*A(x)^(3*n+1)/(3*n+1),
form a table of coefficients in A(x)^(3*n+1) as follows:
A^4: [(1), 4, 22, 188, 2217, 32516, 555972, ...];
A^7: [1, (7), 49, 441, 5131, 73248, 1220457, ...];
A^10: [1, 10, (85), 820, 9590, 134482, 2191060, ...];
A^13: [1, 13, 130, (1352), 16107, 223886, 3582072, ...];
A^16: [1, 16, 184, 2064, (25276), 351072, 5541912, ...];
A^19: [1, 19, 247, 2983, 37772, (527839), 8260174, ...];  ...
in which the main diagonal forms the initial terms of this sequence:
[4/4*(1), 4/7*(7), 4/10*(85), 4/13*(1352), 4/16*(25276), 4/19*(527839), ...].
ALTERNATE GENERATING METHOD.
This sequence forms column zero in the follow array.
Let A denote this sequence, and A^3 the self-convolution cube of A.
Start in row zero with this sequence, A, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A^3 and the remaining terms in a given row to obtain the next row:
[1, 1, 1, 4, 34, 416, 6319, 111124, 2177346, 46440184, 1061938195, ...];
[1, 4, 22, 188, 2217, 32516, 555972, 10655628, 223313220, 5034249556, ...];
[4, 34, 314, 3619, 50829, 833591, 15417781, 313704516, 6900409869, ...];
[34, 416, 5071, 70714, 1131649, 20377616, 404581945, 8712077584, ...];
[416, 6319, 92167, 1472688, 26106282, 508663862, 10756749655, ...];
[6319, 111124, 1843974, 32709364, 630332565, 13142389012, 294045605744, ...];
[111124, 2177346, 39908146, 770233015, 15926231353, 352361228127, ...];
[46440184, 1061938195, 22576531219, 497127293708, 11532665230502, ...]; ...

Crossrefs

Cf. variants: A002293, A145347, A145348, A120974, A145345, A145350, A147664.

Programs

  • PARI
    {a(n)=local(A=1+x,G);for(i=0,n,G=(serreverse(x/(A+x*O(x^n))^3)/x)^(1/3);A=1+x*G^4);polcoeff(A,n)}
  • PARI
    /* This sequence is generated when k=3, m=1: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=3, m=1)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* Prints terms 0..30 */
{A=[1];
for(m=1,30,
  B=A;
  for(i=1,m-1, C=Vec(Ser(A)^3*Ser(B)); B=vector(#C-1,n,C[n+1]) );
  A=concat(A,0);A[#A]=B[1]
);
A} \\ Paul D. Hanna, Jan 10 2016

Formula

G.f.: A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^3) and A(x) = G(x/A(x)^3).
a(n) = [x^(n-1)] 4*A(x)^(3*n+1)/(3*n+1) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 4*A(x)^(3*n+1)/(3*n+1) for n>=1 (see comment).

A383563 G.f. A(x) satisfies A( x*(1+x)/A(x)^2 ) = 1 + x.

Original entry on oeis.org

1, 1, 1, 3, 13, 72, 465, 3362, 26531, 224856, 2024188, 19202830, 190857879, 1978567663, 21319434418, 238109360460, 2750229390071, 32789591062124, 402891169846242, 5094855923807780, 66229610059651788, 884081025776797026, 12107164229698851942, 169954380180177899277, 2443554376412586234247
Offset: 0

Views

Author

Paul D. Hanna, May 26 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 13*x^4 + 72*x^5 + 465*x^6 + 3362*x^7 + 26531*x^8 + 224856*x^9 + 2024188*x^10 + ...
 where A( x*(1+x)/A(x)^2 ) = 1 + x.
RELATED SERIES.
The g.f. of A145345 begins
B(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 203*x^5 + 1398*x^6 + 10706*x^7 + 89120*x^8 + 794347*x^9 + ...
 where B(x/B(x)) = 1 + x*B(x)
 also, B( x/A(x)^2 ) = 1 + x.
The g.f. of A121687 begins
C(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 + 37244*x^7 + 335153*x^8 + 3194510*x^9 + ...
 where C(x) = 1/(1 - x*C(x*C(x))^2)
 also, C( (x/(1+x))/A(x)^2 ) = 1 + x.
C(x)^2 = 1 + 2*x + 7*x^2 + 34*x^3 + 203*x^4 + 1398*x^5 + 10706*x^6 + 89120*x^7 + 794347*x^8 + ...
 where B(x) = 1 + x*C(x)^2.

Links

Crossrefs

Cf. A384265, A145345 (B(x)), A121687 (C(x)).

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoef(x - serreverse(x/Ser(A)) + x*serreverse(x/Ser(A)^2),#A) ); A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = g.f. of A145345 and C(x) = g.f. of A121687 satisfies the following formulas.
(1) A( x*(1+x)/A(x)^2 ) = 1 + x.
(2.a) Series_Reversion( x/A(x) ) = x + x*Series_Reversion( x/A(x)^2 ).
(2.b) [x^n] A(x)^(n+1)/(n+1) = [x^(n-1)] A(x)^(2*n)/n for n >= 1.
(2.c) B(x) = 1 + x*C(x)^2.
(3.a) A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = C(x/B(x)).
(3.b) A(x) = C(x/A(x)^2) where C(x) = A(x*C(x)^2) = B(x*C(x)).
(4.a) A(x) = A(x)^2 - x*C(x/A(x))^2.
(4.b) B( x/A(x)^2 ) = 1 + x.
(4.c) C( (x/(1+x))/A(x)^2 ) = 1 + x.
(4.d) B( (x/(1+x))/A(x)^2 ) = 1 + x*(1+x)/A(x)^2.
(4.e) A( (x/(1+x))/A(x)^2 + x^2/A(x)^4 ) = 1 + x*(1+x)/A(x)^2.
(5.a) A145345(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1) for n >= 0.
(5.b) A121687(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.
(5.c) A145345(n) = [x^(n-1)] C(x)^2 = [x^(n-1)] A(x)^(2*n)/n for n >= 1.

A384265 G.f. A(x) satisfies A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.

Original entry on oeis.org

1, 1, 3, 16, 119, 1087, 11408, 132468, 1663047, 22234598, 313303201, 4618133168, 70815362628, 1124901511837, 18450127411436, 311636597558992, 5409374008212747, 96326565666389514, 1757212245656330130, 32798907320986196010, 625759879697614204041, 12192353855092076824051, 242419156191210763668352
Offset: 0

Views

Author

Paul D. Hanna, May 30 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 119*x^4 + 1087*x^5 + 11408*x^6 + 132468*x^7 + 1663047*x^8 + 22234598*x^9 + 313303201*x^10 + ...
 where A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.
RELATED SERIES.
The g.f. of A145347 begins
B(x) = 1 + x + 4*x^2 + 26*x^3 + 220*x^4 + 2203*x^5 + 24836*x^6 + 306104*x^7 + 4047988*x^8 + ... + A145347(n)*x^n + ...
 where B(x/B(x)) = 1 + x*B(x)^3
 also, B(x) = A( x*B(x) ).
The g.f. of A182954 begins
C(x) = 1 + x + 5*x^2 + 39*x^3 + 381*x^4 + 4284*x^5 + 53163*x^6 + 710810*x^7 + 10085621*x^8 + ... + A182954(n)*x^n + ...
 where C(x) = 1 + x*C(x) * C( x*C(x) )^4
 also, C(x) = A( x*C(x)^2 ).
C(x)^4 = 1 + 4*x + 26*x^2 + 220*x^3 + 2203*x^4 + 24836*x^5 + 306104*x^6 + ...
 where B(x) = 1 + x*C(x)^4.
RELATED TABLE.
The table of coefficients of x^k in A(x)^n begins
  n = 1: [ 1,  1,   3,   16,   119,   1087, ...];
  n = 2: [ 1, (2),  7,   38,   279,   2508, ...];
  n = 3: [ 1,  3, (12),  67,   489,   4338, ...];
  n = 4: [(1), 4,  18, (104),  759,   6664, ...];
  n = 5: [ 1,  5,  25,  150, (1100),  9586, ...];
  n = 6: [ 1, (6), 33,  206,  1524, (13218),...];
  n = 7: [ 1,  7,  42,  273,  2044,  17689, ...];
  n = 8: [ 1,  8, (52), 352,  2674,  23144, ...];
  n = 9: [ 1,  9,  63,  444,  3429,  29745, ...];
  n =10: [ 1, 10,  75, (550), 4325,  37672, ...];
  n =11: [ 1, 11,  88,  671,  5379,  47124, ...];
  n =12: [ 1, 12, 102,  808, (6609), 58320, ...];
  ...
in which the terms in parenthesis illustrate the property
(2.b) [x^n] A(x)^(n+1) = 2 * [x^(n-1)] A(x)^(2*n+2) for n >= 1.
Further, other terms along diagonals obey
(5.a) A145345(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1), and
(5.b) A121687(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.

Links

Crossrefs

Cf. A383563, A145347 (B(x)), A182954 (C(x)).

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoef(x - serreverse(x/Ser(A)) + serreverse(x/Ser(A)^2)^2, #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = g.f. of A145347 and C(x) = g.f. of A182954 satisfy the following formulas.
(1.a) A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.
(1.b) A(x) = A(x)^2 - x*A( sqrt(x - x/A(x)) )^4.
(2.a) x = Series_Reversion( x/A(x) ) - Series_Reversion( x/A(x)^2 )^2.
(2.b) [x^n] A(x)^(n+1) = 2 * [x^(n-1)] A(x)^(2*n+2) for n >= 1.
(2.c) B(x) = 1 + x*C(x)^4 where B(x) = A( x*B(x) ) and C(x) = A( x*C(x)^2 ).
(3.a) A(x) = B( x/A(x) ) where B(x/B(x)) = 1 + x*B(x)^3.
(3.b) A(x) = C( x/A(x)^2 ) where C(x) = 1 + x*C(x) * C( x*C(x) )^4.
(4.a) A(x) = A(x)^2 - x*C( x/A(x) )^4.
(4.b) C(x/A(x)) = A( sqrt(x - x/A(x)) ) = A( x/A(x) * C(x/A(x))^2 ).
(4.c) B(x/A(x)^2) = 1 + x*A(x)^2 = A( x/A(x)^2 * B(x/A(x)^2) ).
(5.a) A145347(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1) for n >= 0.
(5.b) A182954(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.
(5.c) A145347(n) = [x^(n-1)] C(x)^4 = [x^(n-1)] A(x)^(2*n+2)*2/(n+1) for n >= 1.
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