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 1-10 of 10 results.

A373578 Expansion of e.g.f. exp(x * (1 + x^2)^2).

Original entry on oeis.org

1, 1, 1, 13, 49, 241, 2401, 13021, 128353, 1346689, 10615681, 140431501, 1544877841, 17576665393, 264566466529, 3226728670621, 48376006929601, 766753039205761, 11052669865900033, 197019825098096269, 3271213100827557361, 56597110823949654001
Offset: 0

Views

Author

Seiichi Manyama, Jun 10 2024

Keywords

Links

Crossrefs

Cf. A118395, A190863, A373577.
Cf. A361278.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[E^(x*(1 + x^2)^2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 11 2024 *)
  • PARI
    a(n) = n!*sum(k=0, 2*n\5, binomial(2*n-4*k, k)/(n-2*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(2*n/5)} binomial(2*n-4*k,k)/(n-2*k)!.
a(n) == 1 (mod 12).
a(n) = a(n-1) + 6*(n-1)*(n-2)*a(n-3) + 5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5).
a(n) ~ 5^(n/5 - 1/2) * exp(7*5^(-11/5)*n^(1/5) + 2*5^(-3/5)*n^(3/5) - 4*n/5) * n^(4*n/5). - Vaclav Kotesovec, Jun 11 2024

A361567 Expansion of e.g.f. exp(x^2/2 * (1+x)^2).

Original entry on oeis.org

1, 0, 1, 6, 15, 60, 555, 3150, 17745, 158760, 1399545, 10914750, 102920895, 1104323220, 11249313075, 119330961750, 1426411411425, 17429852840400, 213417453474225, 2791671804271350, 38524272522310575, 537569719902715500, 7732658753799054075
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2023

Keywords

Crossrefs

Cf. A047974, A361568, A361569.
Cf. A335344, A361278.

Programs

  • Mathematica
    Table[n! * Sum[Binomial[2*k,n-2*k]/(2^k * k!), {k,0,n/2}], {n,0,20}] (* Vaclav Kotesovec, Mar 25 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/2*(1+x)^2)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/2*sum(j=2, i, j*binomial(2, j-2)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*k,n-2*k)/(2^k * k!).
a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=2..n} k * binomial(2,k-2) * a(n-k)/(n-k)!.
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) = (n-1)*a(n-2) + 3*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(n/4 - 1) * exp(1/128 - 3*2^(-29/4)*n^(1/4) - sqrt(n/2)/16 + 2^(-3/4)*n^(3/4) - 3*n/4) * n^(3*n/4). (End)

A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 19, 25, 1, 1, 1, 9, 37, 97, 81, 1, 1, 1, 11, 61, 241, 581, 331, 1, 1, 1, 13, 91, 481, 1981, 3661, 1303, 1, 1, 1, 15, 127, 841, 4881, 17551, 26335, 5937, 1, 1, 1, 17, 169, 1345, 10001, 55321, 171697, 202049, 26785, 1
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  1,   1,    1,    1,     1, ...
  1,  3,   5,    7,    9,    11, ...
  1,  7,  19,   37,   61,    91, ...
  1, 25,  97,  241,  481,   841, ...
  1, 81, 581, 1981, 4881, 10001, ...

Crossrefs

Columns k=0..4 give A000012, A047974, A361278, A361279, A361280.
Main diagonal gives A361281.
Cf. A293012.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n, binomial(k*j, n-j)/j!);

Formula

E.g.f. of column k: exp(x * (1+x)^k).
T(0,k) = 1; T(n,k) = (n-1)! * Sum_{j=1..n} j * binomial(k,j-1) * T(n-j,k)/(n-j)!.

A377963 Expansion of e.g.f. (1+x) * exp(x*(1+x)^2).

Original entry on oeis.org

1, 2, 7, 34, 173, 1066, 7147, 51962, 412729, 3478258, 31220111, 296409202, 2953487077, 30870965594, 336796018483, 3824230997386, 45114077004017, 551338045973602, 6968344940992279, 90931562913957698, 1222939213021853341, 16929504703420184842, 240909000856701880187
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A018191, A377964.
Cf. A361278, A377965.

Programs

  • PARI
    a(n, s=1, t=2) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} binomial(2*k+1,n-k) / k!.
a(n) = a(n-1) + (4*n-3)*a(n-2) + 3*(n-2)*n*a(n-3) for n > 2.

A373708 Expansion of e.g.f. exp(x * (1 + x^4)^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 241, 1441, 5041, 13441, 393121, 10946881, 99902881, 559025281, 2335441681, 182348406241, 4382526067921, 48882114328321, 355837396998721, 5157802930734721, 312898934463543361, 7129755898022511361, 89524038506304371761, 773103613914955683361
Offset: 0

Views

Author

Seiichi Manyama, Jun 14 2024

Keywords

Crossrefs

Cf. A361278, A373578, A373707.
Cf. A190877, A373524, A373525, A373526.
Cf. A373706.

Programs

  • PARI
    a(n) = n!*sum(k=0, 2*n\9, binomial(2*n-8*k, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(2*n/9)} binomial(2*n-8*k,k)/(n-4*k)!.
a(n) == 1 (mod 240).
a(n) = a(n-1) + 10*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5) + 9*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*a(n-9).

A377965 Expansion of e.g.f. (1+x)^2 * exp(x*(1+x)^2).

Original entry on oeis.org

1, 3, 11, 55, 309, 1931, 13543, 101991, 828425, 7192819, 66002691, 639830423, 6510397501, 69266297595, 768989536799, 8876171274631, 106301772962193, 1318277355041891, 16892429768517115, 223330116792810999, 3041570471301007301, 42611228176879105003
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Cf. A377954, A377966.
Cf. A361278, A377963.
Cf. A343884.

Programs

  • PARI
    a(n, s=2, t=2) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} binomial(2*k+2,n-k) / k!.
From Vaclav Kotesovec, Nov 23 2024: (Start)
Recurrence: (n^2 - 3*n + 4)*a(n) = (n^2 - 3*n + 8)*a(n-1) + 2*(n-1)*(2*n^2 - 5*n + 4)*a(n-2) + 3*(n-2)*(n-1)*(n^2 - n + 2)*a(n-3).
a(n) ~ 3^(n/3 - 7/6) * exp(-4/81 + 3^(-7/3)*n^(1/3) + 2*3^(-2/3)*n^(2/3) - 2*n/3) * n^(2*(n+1)/3) * (1 + 5813*3^(1/3)/(4374*n^(1/3))). (End)

A361570 Expansion of e.g.f. exp( (x * (1+x))^2 ).

Original entry on oeis.org

1, 0, 2, 12, 36, 240, 2280, 15120, 122640, 1330560, 13335840, 136382400, 1657212480, 20860519680, 262278656640, 3585207225600, 52249374777600, 772773281280000, 11907924610982400, 193962388523904000, 3253343368231756800, 56051640629816832000
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2023

Keywords

Crossrefs

Cf. A047974, A361571.
Cf. A052887, A361278.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[(x(1+x))^2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 18 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(1+x))^2)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, i, j*binomial(2, j-2)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*k,n-2*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * binomial(2,k-2) * a(n-k)/(n-k)!.
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) ~ 2^(n/2 - 1) * exp(1/64 - 3*n^(1/4)/2^(13/2) - sqrt(n)/16 + n^(3/4)/sqrt(2) - 3*n/4) * n^(3*n/4).
a(n) = 2*(n-1)*a(n-2) + 6*(n-2)*(n-1)*a(n-3) + 4*(n-3)*(n-2)*(n-1)*a(n-4). (End)

A362774 E.g.f. satisfies A(x) = exp( x * (1+x)^2 * A(x)^2 ).

Original entry on oeis.org

1, 1, 9, 115, 2265, 59701, 1981513, 79441167, 3736418801, 201790517833, 12309193580841, 837132560820139, 62809405894333321, 5154060532188515325, 459202970647825870313, 44146740571635016905991, 4555272678073789024849377, 502153774773932684443210513
Offset: 0

Views

Author

Seiichi Manyama, May 02 2023

Keywords

Links

Crossrefs

Cf. A361278, A362772.
Cf. A052750, A360547, A362776.

Programs

  • Maple
    A362774 := proc(n)
    n!*add((2*k+1)^(k-1) * binomial(2*k,n-k)/k!,k=0..n) ;
end proc:
seq(A362774(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*(1+x)^2)/2)))

Formula

E.g.f.: exp( -LambertW(-2*x * (1+x)^2)/2 ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(2*k,n-k)/k!.

A373707 Expansion of e.g.f. exp(x * (1 + x^3)^2).

Original entry on oeis.org

1, 1, 1, 1, 49, 241, 721, 6721, 124321, 913249, 4243681, 94818241, 1640604241, 14642181841, 131026944049, 3669304504321, 62536989802561, 627395160826561, 10818406189690561, 308036857749752449, 5219006583104930161, 65146235714284117681
Offset: 0

Views

Author

Seiichi Manyama, Jun 14 2024

Keywords

Crossrefs

Cf. A361278, A373578, A373708.
Cf. A190875, A373522, A373523.
Cf. A373680.

Programs

  • PARI
    a(n) = n!*sum(k=0, 2*n\7, binomial(2*n-6*k, k)/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(2*n/7)} binomial(2*n-6*k,k)/(n-3*k)!.
a(n) == 1 (mod 48).
a(n) = a(n-1) + 8*(n-1)*(n-2)*(n-3)*a(n-4) + 7*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*a(n-7).

A373743 Expansion of e.g.f. exp(x^3/6 * (1 + x)^2).

Original entry on oeis.org

1, 0, 0, 1, 8, 20, 10, 280, 3360, 20440, 67200, 462000, 7407400, 73673600, 482081600, 3364761400, 47311264000, 657536880000, 6586994814400, 58707179731200, 740032028736000, 11832726841936000, 161121297104768000, 1857897194273120000, 23875495204536976000
Offset: 0

Views

Author

Seiichi Manyama, Jun 16 2024

Keywords

Crossrefs

Cf. A361568, A372742.
Cf. A361278, A361567.
Cf. A264622.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(2*k,n-3*k)/(6^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(2*k,n-3*k)/(6^k * k!).
a(n) = (n-1)*(n-2)/6 * (3*a(n-3) + 8*(n-3)*a(n-4) + 5*(n-3)*(n-4)*a(n-5)).
Showing 1-10 of 10 results.

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