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.

Previous Showing 11-18 of 18 results.

A377146 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(k,n-2*k)^2.

Original entry on oeis.org

1, 0, 3, 3, 6, 24, 16, 90, 105, 250, 561, 765, 2143, 3108, 6861, 12985, 22221, 47988, 79463, 161451, 293610, 535836, 1042188, 1835898, 3534766, 6399198, 11805756, 22021232, 39718497, 74193924, 134489713, 247165839, 453235266, 822748406, 1512078192, 2741606052
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2024

Keywords

Crossrefs

Cf. A377145, A377147.
Cf. A298567, A375218.
Cf. A089627.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(k+2, 2)*binomial(k, n-2*k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=2, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

Formula

G.f.: ((1-x^2-x^3)^2 + 2*x^5) / ((1-x^2-x^3)^2 - 4*x^5)^(5/2).

A377150 a(n) = Sum_{k=0..floor(n/3)} binomial(k+3,3) * binomial(k,n-3*k)^2.

Original entry on oeis.org

1, 0, 0, 4, 4, 0, 10, 40, 10, 20, 180, 180, 55, 560, 1260, 616, 1435, 5600, 5684, 4424, 18956, 33720, 24780, 55944, 147249, 157560, 182280, 523540, 826440, 802560, 1681966, 3531880, 4072035, 5671084, 12941764, 19281064, 22523175, 43823520, 80254746, 99744776
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2024

Keywords

Links

Crossrefs

Cf. A377148, A377149.
Cf. A089627.

Programs

  • Maple
    f:= proc(n) local k; add(binomial(k+3,3)*binomial(k,n-3*k)^2,k=0..floor(n/3)) end proc:
map(f, [$0..50]); # Robert Israel, Dec 05 2024
  • PARI
    a(n) = sum(k=0, n\3, binomial(k+3, 3)*binomial(k, n-3*k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=3, M=40, x='x+O('x^M), X=1-x^3-x^4, Y=7); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

Formula

G.f.: (1-x^3-x^4) * ((1-x^3-x^4)^2 + 6*x^7) / ((1-x^3-x^4)^2 - 4*x^7)^(7/2).

A194586 Triangle read by rows, T(n,k) the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*(k mod 2)*q^k.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 3, 0, 6, 0, 4, 0, 24, 0, 0, 5, 0, 60, 0, 30, 0, 6, 0, 120, 0, 180, 0, 0, 7, 0, 210, 0, 630, 0, 140, 0, 8, 0, 336, 0, 1680, 0, 1120, 0, 0, 9, 0, 504, 0, 3780, 0, 5040, 0, 630, 0, 10, 0, 720, 0, 7560, 0, 16800, 0, 6300, 0, 0, 11, 0, 990, 0, 13860, 0, 46200, 0, 34650, 0, 2772, 0, 12
Offset: 0

Views

Author

Peter Luschny, Aug 29 2011

Keywords

Comments

Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the complementary Motzkin numbers A005717. (See A089627 for the Motzkin numbers and A163649 for the extended Motzkin numbers.)

Examples

			               0
              0, 1
            0, 2, 0
           0, 3, 0, 6
         0, 4, 0, 24, 0
       0, 5, 0, 60, 0, 30
    0, 6, 0, 120, 0, 180, 0
  0, 7, 0, 210, 0, 630, 0, 140
                0
                q
               2 q
            3 q + 6 q^3
           4 q + 24 q^3
       5 q + 60 q^3  + 30 q^5
      6 q + 120 q^3  + 180 q^5
  7 q + 210 q^3  + 630 q^5  + 140 q^7

Links

Crossrefs

Row sums are A109188. Cf. A056040, A005717, A163649, A089627.

Programs

  • Maple
    A194586 := proc(n,k) local j, swing; swing := n -> n!/iquo(n,2)!^2:
add(binomial(n,j)*swing(j)*q^j*(j mod 2),j=0..n); coeff(%,q,k) end:
seq(print(seq(A194586(n,k),k=0..n)),n=0..8);
  • Mathematica
    sf[n_] := n!/Quotient[n, 2]!^2;
row[n_] := Sum[Binomial[n, j] sf[j] q^j Mod[j, 2], {j, 0, n}] // CoefficientList[#, q]& // PadRight[#, n+1]&;
Table[row[n], {n, 0, 12}] (* Jean-François Alcover, Jun 26 2019 *)

Formula

egf(x,y) = x*y*exp(x)*BesselI(0,2*x*y).

A377147 a(n) = Sum_{k=0..floor(n/3)} binomial(k+2,2) * binomial(k,n-3*k)^2.

Original entry on oeis.org

1, 0, 0, 3, 3, 0, 6, 24, 6, 10, 90, 90, 25, 240, 540, 261, 540, 2100, 2128, 1533, 6321, 11236, 8064, 16884, 44173, 46980, 51156, 142939, 224991, 212400, 423426, 882660, 1006875, 1338558, 2991318, 4431669, 5034296, 9457704, 17178678, 21059737, 30809286, 59843394, 86518266
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2024

Keywords

Crossrefs

Cf. A377145, A377146.
Cf. A375470, A376721.
Cf. A089627.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(k+2, 2)*binomial(k, n-3*k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=2, M=50, x='x+O('x^M), X=1-x^3-x^4, Y=7); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

Formula

G.f.: ((1-x^3-x^4)^2 + 2*x^7) / ((1-x^3-x^4)^2 - 4*x^7)^(5/2).

A377149 a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(k,n-2*k)^2.

Original entry on oeis.org

1, 0, 4, 4, 10, 40, 30, 180, 215, 580, 1316, 1960, 5719, 8624, 20420, 39536, 71985, 160584, 276664, 588180, 1099786, 2098480, 4201594, 7665724, 15266640, 28422684, 54252560, 103928876, 193166861, 371012360, 690296162, 1304353740, 2450895828, 4565652908
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2024

Keywords

Crossrefs

Cf. A377148, A377150.
Cf. A089627.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(k+3, 3)*binomial(k, n-2*k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=3, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

Formula

G.f.: (1-x^2-x^3) * ((1-x^2-x^3)^2 + 6*x^5) / ((1-x^2-x^3)^2 - 4*x^5)^(7/2).

A382841 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^2.

Original entry on oeis.org

1, 1, 5, 37, 181, 1301, 9401, 65465, 498037, 3796021, 29221705, 230396585, 1828448425, 14651160265, 118544522045, 965075143037, 7907605360757, 65162569952245, 539515760866889, 4486877961224297, 37463151704756281, 313909383754331801, 2638892573249746445, 22249830926517611917
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 06 2025

Keywords

Comments

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - w) - (x*y)^2*z*w).

Links

Crossrefs

Cf. A000984, A002426, A005259, A005260, A051286, A181546, A275027, A277247, A382842.
Cf. A089627.

Programs

  • Maple
    a:= n-> add(combinat[multinomial](n, n-2*k, k$2)^2, k=0..n/2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 07 2025
  • Mathematica
    Table[Sum[(Binomial[n, k] Binomial[n - k, k])^2, {k, 0, Floor[n/2]}], {n, 0, 23}]
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1, 1, 1}, 16], {n, 0, 23}]
Table[SeriesCoefficient[1/((1 - x) (1 - y) (1 - z) (1 - w) - (x y)^2 z w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {w, 0, n}], {n, 0, 23}]

Formula

a(n) ~ 3^(2*n+2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 07 2025
a(n) = Sum_{k=0..floor(n/2)} A089627(n,k)^2. - Alois P. Heinz, Apr 07 2025

A382842 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^3.

Original entry on oeis.org

1, 1, 9, 217, 1945, 35001, 764001, 12079089, 250222617, 5424133465, 107360983009, 2358751625649, 52540471866961, 1147794435985393, 26151265459123065, 600227875293254217, 13779170435209475097, 322302377797126709913, 7582484532013652243169, 179184911648568670363185, 4275721755296040840336945
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 06 2025

Keywords

Comments

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - u)*(1 - v)*(1 - w) - (x*y*z)^2*u*v*w).

Crossrefs

Cf. A000172, A002426, A069865, A092813, A181545, A382841.
Cf. A089627.

Programs

  • Maple
    a:= n-> add(combinat[multinomial](n, n-2*k, k$2)^3, k=0..n/2):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 07 2025
  • Mathematica
    Table[Sum[(Binomial[n, k] Binomial[n - k, k])^3, {k, 0, Floor[n/2]}], {n, 0, 20}]
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, 1/2 - n/2, -n/2, -n/2, -n/2}, {1, 1, 1, 1, 1}, 64], {n, 0, 20}]
Table[SeriesCoefficient[1/((1 - x) (1 - y) (1 - z) (1 - u) (1 - v) (1 - w) - (x y z)^2 u v w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {u, 0, n}, {v, 0, n}, {w, 0, n}], {n, 0, 20}]

Formula

a(n) ~ 3^(3*n+3) / (8 * Pi^(5/2) * n^(5/2)). - Vaclav Kotesovec, Apr 07 2025
a(n) = Sum_{k=0..floor(n/2)} A089627(n,k)^3. - Alois P. Heinz, Apr 07 2025

A229032 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).

Original entry on oeis.org

1, 2, 0, 3, 4, 0, 4, 16, 0, 0, 5, 40, 16, 0, 0, 6, 80, 96, 0, 0, 0, 7, 140, 336, 64, 0, 0, 0, 8, 224, 896, 512, 0, 0, 0, 0, 9, 336, 2016, 2304, 256, 0, 0, 0, 0, 10, 480, 4032, 7680, 2560, 0, 0, 0, 0, 0, 11, 660, 7392, 21120, 14080, 1024, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Rui Duarte and António Guedes de Oliveira, Sep 11 2013

Keywords

Comments

Row n is the sum of the convolution of A089627(p,i) with A089627(n-p,i), for p=0,1,...,n.

Examples

			Triangle:
1
2, 0
3, 4, 0
4, 16, 0, 0
5, 40, 16, 0, 0
6, 80, 96, 0, 0, 0
7, 140, 336, 64, 0, 0, 0
8, 224, 896, 512, 0, 0, 0, 0
9, 336, 2016, 2304, 256, 0, 0, 0, 0
10, 480, 4032, 7680, 2560, 0, 0, 0, 0, 0
11, 660, 7392, 21120, 14080, 1024, 0, 0, 0, 0, 0

Links

Formula

T(n,k) = 4^k * C(n+1, 2*k+1).
T(n,k) = sum(p=0..n, sum(i=0..k, C(p,i)*C(p-i, i)*C(n-p,k-i)*C(n-p-k+i, k-i))).
T(n,k) = A085841(n/2,k), if n is even.
T(n,k) = 2^k * A105070(n,k).
Previous Showing 11-18 of 18 results.

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