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 1-8 of 8 results.

A133316 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n/(1+x)^(n^2).

Original entry on oeis.org

1, 1, 2, 8, 54, 544, 7508, 133704, 2943194, 77589536, 2391477804, 84582890704, 3382005372970, 151034046369696, 7458091839548356, 403808650013237224, 23801728042233670770, 1517930142778063770304, 104179592763803229618620
Offset: 0

Views

Author

Paul D. Hanna, Dec 24 2010

Keywords

Examples

			1/(1-x) = 1 + x/(1+x) + 2*x^2/(1+x)^4 + 8*x^3/(1+x)^9 + 54*x^4/(1+x)^16 +...

Crossrefs

Cf. A141761, A177447, A215241, A215242, A215243.

Programs

  • PARI
    {a(n)=if(n==0,1,polcoeff(-(1-x)*sum(m=0,n-1,a(m)*x^m/(1+x +x*O(x^n))^(m^2)),n))}
  • PARI
    {a(n)=if(n==0, 1, 1 - sum(j=0, n-1, a(j)*(-1)^(n-j)*binomial(j^2+n-j-1, n-j)))}

Formula

a(n) = 1 - Sum_{j=0..n-1} a(j) * (-1)^(n-j) * C(j^2 + n-j-1, n-j) for n>0, with a(0)=1.

A141760 Triangle T, read by rows, where the g.f. of column k in matrix power T^m is given by: 1/(1-x)^m = Sum_{n>=k} [T^m](n,k) * x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2} for k>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 6, 3, 1, 1, 26, 26, 13, 4, 1, 1, 154, 154, 77, 23, 5, 1, 1, 1188, 1188, 594, 175, 36, 6, 1, 1, 11474, 11474, 5737, 1678, 336, 52, 7, 1, 1, 134432, 134432, 67216, 19579, 3863, 576, 71, 8, 1, 1, 1863168, 1863168, 931584, 270683, 52944, 7731
Offset: 0

Views

Author

Paul D. Hanna, Jul 18 2008

Keywords

Examples

			Triangle T begins:
1;
1, 1;
1, 1, 1;
2, 2, 1, 1;
6, 6, 3, 1, 1;
26, 26, 13, 4, 1, 1;
154, 154, 77, 23, 5, 1, 1;
1188, 1188, 594, 175, 36, 6, 1, 1;
11474, 11474, 5737, 1678, 336, 52, 7, 1, 1;
134432, 134432, 67216, 19579, 3863, 576, 71, 8, 1, 1;
1863168, 1863168, 931584, 270683, 52944, 7731, 911, 93, 9, 1, 1; ...
Matrix square, T^2, begins:
1;
2, 1;
3, 2, 1;
7, 5, 2, 1;
23, 17, 7, 2, 1;
105, 79, 33, 9, 2, 1;
641, 487, 205, 55, 11, 2, 1;
5034, 3846, 1626, 433, 83, 13, 2, 1; ...
where g.f. for column k of matrix square T^2 is:
1/(1-x)^2 = Sum_{n>=0} [T^2](n,k)*x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2}.
Matrix inverse, T^-1, begins:
1;
-1, 1;
0, -1, 1;
0, -1, -1, 1;
0, -2, -2, -1, 1;
0, -7, -7, -3, -1, 1;
0, -37, -37, -15, -4, -1, 1;
0, -268, -268, -106, -26, -5, -1, 1; ...
Let U = unsigned T^-1 with leftmost column dropped,
then U = A107876 where [U^k](n,k) = U(n,k-1) for n>=k>0.
The g.f. for column k of matrix inverse T^-1 is:
1-x = Sum_{n>=0} [T^-1](n,k) * x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2}.
MATRIX PRODUCTS:
T = P(1)^-1 * P(2) = P(2)^-1 * P(3) = P(m)^-1 * P(m+1);
P(1) begins:
1;
1, 1;
2, 2, 1;
8, 7, 3, 1;
57, 42, 16, 4, 1;
638, 386, 130, 29, 5, 1;
9949, 4944, 1471, 299, 46, 6, 1; ...
where [P(1)](n,k) = [x^(n-k)] 1/(1-x)*(1+x)^{n(n-1)/2-k(k-1)/2};
P(2) begins:
1;
2, 1;
5, 3, 1;
20, 12, 4, 1;
129, 72, 23, 5, 1;
1268, 630, 187, 38, 6, 1;
17548, 7599, 2063, 392, 57, 7, 1; ...
where [P(2)](n,k) = [x^(n-k)] 1/(1-x)^2*(1+x)^{n(n-1)/2-k(k-1)/2};
P(3) begins:
1;
3, 1;
9, 4, 1;
38, 18, 5, 1;
240, 111, 31, 6, 1;
2223, 955, 256, 48, 7, 1;
28672, 11124, 2794, 500, 69, 8, 1; ...
where [P(3)](n,k) = [x^(n-k)] 1/(1-x)^3*(1+x)^{n(n-1)/2-k(k-1)/2}.

Crossrefs

Cf. columns: A141761, A141762, A141763; A107876 (unsigned inverse).

Programs

  • Mathematica
    T[n_, k_, m_] := T[n, k, m] = If[nJean-François Alcover, Sep 19 2016, adapted from PARI *)
  • PARI
    T(n,k,m=1)=if(n

Formula

Matrix powers satisfy: T^m = P(i)^-1 * P(m+i) for all m and i, where P(m) is given by:
[P(m)](n,k) = [x^(n-k)] 1/(1-x)^m * (1+x)^{n(n-1)/2 - k(k-1)/2} for n>=k>=0.
Let U = unsigned matrix inverse (T^-1) with leftmost column dropped, then U = A107876 where [U^k](n,k) = U(n,k-1) for n>=k>0.
G.f. for column k of T: 1/(1-x) = Sum_{n>=0} T(n,k)*x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2}.
T(n,k) = 1 - Sum_{j=k..n-1} T(j,k)*(-1)^(n-j)*C(j(j-1)/2 - k(k-1)/2 + n-j-1, n-j) for n>k with T(k,k)=1 for k>=0.
G.f. for column k of matrix power T^m:
1/(1-x)^m = Sum_{n>=0} [T^m](n,k)*x^(n-k)/(1+x)^{n*(n-1)/2 - k*(k-1)/2}.
[T^m](n,k) = C(m+n-1,n) - Sum_{j=k..n-1} [T^m](j,k)*(-1)^(n-j)*C(j(j-1)/2 - k(k-1)/2 + n-j-1,n-j) for n>k with [T^m](k,k)=1 for k>=0.

A141762 Column 2 of triangle A141760.

Original entry on oeis.org

1, 1, 3, 13, 77, 594, 5737, 67216, 931584, 14968423, 274312910, 5657512947, 129866646887, 3287152235160, 91025011377693, 2738909774003719, 89027345548731677, 3110096516555803509, 116244489639439112395
Offset: 0

Views

Author

Paul D. Hanna, Jul 18 2008

Keywords

Crossrefs

Cf. A141760, A141761, A141763, A141764.

Programs

  • PARI
    {a(n)=if(n==0,1,1 - sum(j=0,n-1,a(j)*(-1)^(n-j)*binomial((j+1)*(j+2)/2-1+n-j-1,n-j)))}

Formula

G.f.: 1/(1-x) = Sum_{n>=0} a(n) * x^n/(1+x)^[(n+1)*(n+2)/2 - 1].
a(n) = 1 - Sum_{j=0..n-1} a(j) * (-1)^(n-j) * C((j+1)(j+2)/2 + n-j-2, n-j) for n>0, with a(0)=1.

A141763 Column 3 of triangle A141760.

Original entry on oeis.org

1, 1, 4, 23, 175, 1678, 19579, 270683, 4342447, 79498622, 1638471038, 37592670383, 951214496814, 26333793485772, 792232525678756, 25747819699179668, 899388184082559576, 33613386298645020835, 1338749843351681925409
Offset: 0

Views

Author

Paul D. Hanna, Jul 18 2008

Keywords

Crossrefs

Cf. A141760, A141761, A141762, A141764.

Programs

  • PARI
    {a(n)=if(n==0,1,1 - sum(j=0,n-1,a(j)*(-1)^(n-j)*binomial((j+2)*(j+3)/2-3+n-j-1,n-j)))}

Formula

G.f.: 1/(1-x) = Sum_{n>=0} a(n) * x^n/(1+x)^{(n+2)*(n+3)/2 - 3}.
a(n) = 1 - Sum_{j=0..n-1} a(j) * (-1)^(n-j) * C((j+2)(j+3)/2 + n-j-4, n-j) for n>0, with a(0)=1.

A141764 Column 4 of triangle A141760.

Original entry on oeis.org

1, 1, 5, 36, 336, 3863, 52944, 844777, 15411547, 316888806, 7258887646, 183464090374, 5074934980977, 152584140132667, 4956824851572271, 173086933804868142, 6467223731119047195, 257524741272166890188
Offset: 0

Views

Author

Paul D. Hanna, Jul 18 2008

Keywords

Crossrefs

Cf. A141760, A141761, A141762, A141763.

Programs

  • PARI
    {a(n)=if(n==0,1,1 - sum(j=0,n-1,a(j)*(-1)^(n-j)*binomial((j+3)*(j+4)/2-6+n-j-1,n-j)))}

Formula

G.f.: 1/(1-x) = Sum_{n>=0} a(n) * x^n/(1+x)^{(n+3)*(n+4)/2 - 6}.
a(n) = 1 - Sum_{j=0..n-1} a(j) * (-1)^(n-j) * C((j+3)(j+4)/2 + n-j-7, n-j) for n>0, with a(0)=1.

A182951 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n).

Original entry on oeis.org

1, 1, 3, 16, 134, 1596, 25193, 501236, 12118038, 346373740, 11460810227, 431732603292, 18269225018646, 858920382899880, 44455946598501069, 2513531512113074244, 154218539815668325502, 10209332972405039928876
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2010

Keywords

Examples

			1/(1-x) = 1 + x/(1+x)^2 + 3*x^2/(1+x)^6 + 16*x^3/(1+x)^12 + 134*x^4/(1+x)^20 +...

Crossrefs

Cf. A182952, A133316, A141761.

Programs

  • PARI
    {a(n)=if(n==0,1,polcoeff(-(1-x)*sum(m=0,n-1,a(m)*x^m/(1+x +x*O(x^n))^(m*(m+1))),n))}
  • PARI
    {a(n)=if(n==0, 1, 1 - sum(j=0, n-1, a(j)*(-1)^(n-j)*binomial(j*(j+1)+n-j-1, n-j)))}

Formula

a(n) = 1 - Sum_{k=0..n-1} a(k)*(-1)^(n-k)*C(k^2+k + n-k-1, n-k) for n>0, with a(0)=1.

A182952 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n/(1+x)^(n*(3*n-1)/2).

Original entry on oeis.org

1, 1, 2, 10, 92, 1314, 26216, 682006, 22067246, 858473488, 39151350362, 2052833191416, 121860108702876, 8088426308992214, 594165066779656784, 47891997458689633520, 4205027188507582359156, 399677541092136186656238
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2010

Keywords

Examples

			1/(1-x) = 1 + x/(1+x) + 2*x^2/(1+x)^5 + 10*x^3/(1+x)^12 + 92*x^4/(1+x)^22 + 1314*x^5/(1+x)^35 + 26216*x^6/(1+x)^51 +...

Crossrefs

Cf. A133316, A182951, A182956, A141761.

Programs

  • PARI
    {a(n)=if(n==0,1,polcoeff(-(1-x)*sum(m=0,n-1,a(m)*x^m/(1+x +x*O(x^n))^(m*(3*m-1)/2)),n))}
  • PARI
    {a(n)=if(n==0, 1, 1 - sum(j=0, n-1, a(j)*(-1)^(n-j)*binomial(j*(3*j-1)/2+n-j-1, n-j)))}

Formula

a(n) = 1 - Sum_{k=0..n-1} a(k)*(-1)^(n-k)*C(k(3k-1)/2 + n-k-1, n-k) for n>0, with a(0)=1.

A182956 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n/(1+x)^(n*(3*n+1)/2).

Original entry on oeis.org

1, 1, 3, 19, 206, 3324, 72951, 2050623, 70794951, 2911448386, 139376166446, 7628685374172, 470631647696157, 32346417958899335, 2452988261647043436, 203594274671070109776, 18366854200080039470784, 1790264247095540545539321
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2010

Keywords

Examples

			1/(1-x) = 1 + x/(1+x)^2 + 3*x^2/(1+x)^7 + 19*x^3/(1+x)^15 + 206*x^4/(1+x)^26 + 3324*x^5/(1+x)^40 + 72951*x^6/(1+x)^57 +...

Links

Crossrefs

Cf. A133316, A182951, A182952, A141761.

Programs

  • Mathematica
    nmax=20; b=ConstantArray[0,nmax+1]; b[[1]]=1; Do[b[[n+1]]=1-Sum[b[[j+1]]*(-1)^(n-j)*Binomial[j*(3*j+1)/2+n-j-1,n-j],{j,0,n-1}];,{n,1,nmax}]; b  (* Vaclav Kotesovec, Jul 22 2014 *)
  • PARI
    {a(n)=if(n==0,1,polcoeff(-(1-x)*sum(m=0,n-1,a(m)*x^m/(1+x +x*O(x^n))^(m*(3*m+1)/2)),n))}
  • PARI
    {a(n)=if(n==0, 1, 1 - sum(j=0, n-1, a(j)*(-1)^(n-j)*binomial(j*(3*j+1)/2+n-j-1, n-j)))}

Formula

a(n) = 1 - Sum_{k=0..n-1} a(k)*(-1)^(n-k)*C(k(3k+1)/2 + n-k-1, n-k) for n>0, with a(0)=1.
Showing 1-8 of 8 results.

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