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 1-6 of 6 results.

A128552 Column 2 of triangle A128545; a(n) is the coefficient of q^(2n+4) in the central q-binomial coefficient [2n+4,n+2].

Original entry on oeis.org

1, 3, 8, 18, 39, 75, 141, 251, 433, 724, 1185, 1892, 2972, 4588, 6981, 10480, 15553, 22821, 33164, 47746, 68163, 96542, 135747, 189550, 262997, 362691, 497339, 678300, 920417, 1242898, 1670688, 2235880, 2979809, 3955422, 5230471, 6891234
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2007

Keywords

Comments

Column 1 of triangle A128552 equals the partitions of n (A000041).
a(n) is the number of partitions of the integer 2n+4 into at most n+2 summands each of which is at most n+2. - Geoffrey Critzer, Sep 27 2013

Examples

			a(2) = 8 because we have: 4+4 = 4+3+1 = 4+2+2 = 4+2+1+1 = 3+3+2 = 3+3+1+1 = 3+2+2+1 = 2+2+2+2. - _Geoffrey Critzer_, Sep 27 2013

Links

Crossrefs

Cf. A128545; A128553, A128554.

Programs

  • Maple
    with(combinat): p:= numbpart:
s:= proc(n) s(n):= p(n) +`if`(n>0, s(n-1), 0) end:
a:= n-> p(2*n+4) -2*s(n+1):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 27 2013
  • Mathematica
    Table[nn=2n;Coefficient[Series[Product[(1-x^(n+i))/(1-x^i),{i,1,n}],{x,0,nn}],x^(2n)],{n,1,37}] (* Geoffrey Critzer, Sep 27 2013 *)
  • PARI
    {a(n)=polcoeff(prod(j=n+3,2*n+4,1-q^j)/prod(j=1,n+2,1-q^j),2*n+4,q)}
  • PARI
    {a(n)=numbpart(2*n+4)-2*sum(k=0,n+1,numbpart(k))} \\ Paul D. Hanna, Feb 06 2013

Formula

a(n) = A000041(2*n+4) - 2*Sum_{k=0..n+1} A000041(k), where A000041(n) = number of partitions of n, due to a formula given in the Fu and Sellers paper. - Paul D. Hanna, Feb 06 2013

A128553 Column 3 of triangle A128545; a(n) is the coefficient of q^(3n+9) in the central q-binomial coefficient [2n+6,n+3].

Original entry on oeis.org

1, 5, 18, 58, 155, 383, 867, 1860, 3782, 7409, 13989, 25683, 45905, 80262, 137453, 231269, 382689, 624061, 1003787, 1594796, 2504587, 3892100, 5988403, 9129707, 13798792, 20688367, 30781988, 45473815, 66723042, 97276842, 140959603
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2007

Keywords

Crossrefs

Cf. A128545; A128552, A128554.

Programs

  • PARI
    a(n)=polcoeff(prod(j=n+4,2*n+6,1-q^j)/prod(j=1,n+3,1-q^j),3*n+9,q)

A128554 Column 4 of triangle A128545; a(n) is the coefficient of q^(4n+16) in the central q-binomial coefficient [2n+8,n+4].

Original entry on oeis.org

1, 7, 39, 155, 526, 1555, 4192, 10465, 24620, 55038, 117966, 243723, 487842, 949446, 1802547, 3346632, 6089910, 10881277, 19121293, 33091141, 56466398, 95105255, 158256685, 260386761, 423932473, 683409993, 1091521679, 1728156294
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2007

Keywords

Crossrefs

Cf. A128545; A128552, A128553.

Programs

  • PARI
    a(n)=polcoeff(prod(j=n+5,2*n+8,1-q^j)/prod(j=1,n+4,1-q^j),4*n+16,q)

A123610 Triangle read by rows, where T(n,k) = (1/n)*Sum_{d|(n,k)} phi(d) * binomial(n/d,k/d)^2 for n >= k > 0, with T(n,0) = 1 for n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 20, 20, 5, 1, 1, 6, 39, 68, 39, 6, 1, 1, 7, 63, 175, 175, 63, 7, 1, 1, 8, 100, 392, 618, 392, 100, 8, 1, 1, 9, 144, 786, 1764, 1764, 786, 144, 9, 1, 1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1, 1, 11, 275, 2475, 9900
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2006

Keywords

Comments

A variant of the triangle A047996 of circular binomial coefficients.

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  4,  10,    4,    1;
  1,  5,  20,   20,    5,    1;
  1,  6,  39,   68,   39,    6,    1;
  1,  7,  63,  175,  175,   63,    7,    1;
  1,  8, 100,  392,  618,  392,  100,    8,   1;
  1,  9, 144,  786, 1764, 1764,  786,  144,   9,  1;
  1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1;
  ...
Example of column g.f.s are:
column 1: 1/(1 - x)^2;
column 2: Ser([1, 1, 3, 1]) / ((1 - x)^2*(1 - x^2)^2) = g.f. of A005997;
column 3: Ser([1, 2, 11, 26, 30, 26, 17, 6, 1]) / ((1 - x)^2*(1 - x^2)^2*(1 -x^3)^2);
column 4: Ser([1, 3, 28, 94, 240, 440, 679, 839, 887, 757, 550, 314, 148, 48, 11, 1]) / ((1 - x)^2*(1 - x^2)^2*(1 - x^3)^2*(1 - x^4)^2);
where Ser() denotes a polynomial in x with the given coefficients, as in Ser([1, 1, 3, 1]) = (1 + x + 3*x^2 + x^3).

Links

Crossrefs

Cf. Columns: A005997, A123613, A123614, A123615, A123616.
Cf. A123611 (row sums), A123612 (antidiagonal sums), A123617 (central terms).
Cf. A123618, A123619, A047996 (variant), A128545.

Programs

  • Mathematica
    T[, 0] = 1; T[n, k_] := 1/n DivisorSum[n, If[GCD[k, #] == #, EulerPhi[#]* Binomial[n/#, k/#]^2, 0]&]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)
  • PARI
    {T(n,k)=if(k==0,1,(1/n)*sumdiv(n,d,if(gcd(k,d)==d, eulerphi(d)*binomial(n/d,k/d)^2,0)))}

Formula

T(2*n+1, n) = (2*n + 1)*A000108(n)^2 = (2*n + 1)*((2*n)!/(n!(n+1)!))^2 = A000891(n) for n >= 0.
Row sums are 2*A047996(2*n,n) = 2*A003239(n) for n > 0.
Row sums equal the row sums of triangle A128545.
For n >= 1, the g.f. of column n has the form: P_n(x)/(Product_{m=1..n} (1 - x^m)^2), where P_n(x) is a polynomial with n^2 coefficients such that the sum of the coefficients is P_n(1) = (2*n - 1)!.
From Petros Hadjicostas, Oct 24 2017: (Start)
Proofs of the following formulae can be found in the links.
G.f.: Sum_{n>=1, k>=0} T(n,k)*x^n*y^k = -Sum_{s>=1} (phi(s)/s)*log(f(x^s,y^s)), where phi(s) is Euler's totient function at s, f(x,y) = (sqrt(g(x,y)) + 1 -(1 + y)*x)/2, and g(x,y) = 1 - 2*(1 + y)*x + (1 - y)^2*x^2. (Term T(0,0) is not used in this g.f.)
Row g.f.: Sum_{k>=0} T(n,k)*y^k = (1/n)*Sum_{d|n} phi(d)*R(n/d, y^d), where R(m, y) = [z^m] (1 + (1 + y)*z + y*z^2)^m. (End)

A123611 Row sums of triangle A123610.

Original entry on oeis.org

1, 2, 4, 8, 20, 52, 160, 492, 1620, 5408, 18504, 64132, 225440, 800048, 2865720, 10341208, 37568340, 137270956, 504176992, 1860277044, 6892335720, 25631327688, 95640894056, 357975249028, 1343650267296, 5056424257552
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2006

Keywords

Links

Crossrefs

Cf. A047996, A003239; A123610 (triangle), A123612 (antidiagonal sums); central terms: A123617, A123618.
Cf. A128545.

Programs

  • Mathematica
    Total /@ Table[If[k == 0, 1, 1/n DivisorSum[n, If[Divisible[k, #], EulerPhi[#] Binomial[n/#, k/#]^2, 0] &]], {n, 0, 25}, {k, 0, n}] (* Michael De Vlieger, Apr 03 2017, after Jean-François Alcover at A123610 *)
  • PARI
    {a(n)=sum(k=0,n,if(k==0,1,(1/n)*sumdiv(n,d,if(gcd(k,d)==d, eulerphi(d)*binomial(n/d,k/d)^2,0))))}

Formula

a(n) = 2*A047996(2*n,n) = 2*A003239(n) for n > 0, where A047996 is the triangle of circular binomial coefficients and A003239(n) = number of rooted planar trees with n non-root nodes.
Also equals the row sums of triangle A128545, where A128545(n,k) is the coefficient of q^(n*k) in the q-binomial coefficient [2n,n] for n >= k >= 0.
a(n) = (1/n) * Sum_{d | n} phi(n/d) * binomial(2*d, d) for n>0. - Andrew Howroyd, Apr 02 2017
G.f.: 1 - Sum_{n>=1} (phi(n)/n) * log((1-2*x^n + sqrt(1-4*x^n))/2) = 1 - 2*Sum_{n>=1} (phi(n)/n) * log((1+sqrt(1-4*x^n))/2). (Except for the term a(0) = 1, the first g.f. follows from the g.f. in A123610 by setting y=1, as suggested by P. D. Hanna.) - Petros Hadjicostas, Oct 24 2017

A128562 Triangle, read by rows, where T(n,k) is the coefficient of q^((n+1)*k) in the q-binomial coefficient [2*n+1, n] for n >= k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 12, 6, 1, 1, 10, 29, 29, 10, 1, 1, 14, 61, 94, 61, 14, 1, 1, 21, 120, 263, 263, 120, 21, 1, 1, 29, 222, 645, 910, 645, 222, 29, 1, 1, 41, 392, 1468, 2724, 2724, 1468, 392, 41, 1, 1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2007

Keywords

Comments

Row sums equal a shifted version of A003239 (number of rooted planar trees with n non-root nodes). Column 1 is a shifted version of A000065 (-1 + number of partitions of n). Column 2 is a shifted version of A128563. This array is a variant of triangles A128545 and A047812 (Parker's partition triangle).

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   4,    1;
  1,  6,  12,    6,    1;
  1, 10,  29,   29,   10,    1;
  1, 14,  61,   94,   61,   14,    1;
  1, 21, 120,  263,  263,  120,   21,    1;
  1, 29, 222,  645,  910,  645,  222,   29,   1;
  1, 41, 392, 1468, 2724, 2724, 1468,  392,  41,  1;
  1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1;
  ...

Crossrefs

Cf. A000065 (column 1), A003239 (row sums), A128563 (column 2).
Variants are A047812 and A128545.

Programs

  • PARI
    T(n,k)=if(n

Formula

T(n,k) = [q^((n+1)*k)] Product_{j=n+1..2*n+1}(1-q^j) / Product_{j=1..n+1}(1-q^j).

Extensions

Minor edits by Petros Hadjicostas, Jun 01 2020
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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