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 101-107 of 107 results.

A276964 a(n) = A000262(n)*A000262(n+1).

Original entry on oeis.org

1, 3, 39, 949, 36573, 2029551, 152451283, 14840686449, 1812664465209, 270925848659323, 48571769994336831, 10276325760127883853, 2531148652596607988629, 717525135328209346300839, 231804543407519025287933163
Offset: 0

Views

Author

Emanuele Munarini, Sep 27 2016

Keywords

Crossrefs

Cf. A000262, A105278, A008297, A276960.

Programs

  • Mathematica
    Table[HypergeometricPFQ[{-n+1,-n},{},1]HypergeometricPFQ[{-n,-n-1},{},1],{n,0,100}]
  • Maxima
    makelist(hypergeometric([-n+1,-n],[],1)*hypergeometric([-n,-n-1],[],1),n,0,12);

Formula

Recurrence: (2*n+3)*a(n+3)-(2*n+3)*(3*n^2+19*n+29)*a(n+2)+(n+2)*(n+1)*(2*n+7)*(3*n^2+13*n+13)*a(n+1)-n*(n+1)^3*(n+2)^2*(2*n+7)*a(n) = 0.
Asymptotic: a(n) ~ (1/2)*exp(-2*n+2*sqrt(n)+2*sqrt(n+1)-1)*(475/(110592*n^(3/2))+9025/(21233664*n^2)-5/(24*sqrt(n))-35/(1152*n)+1)*n^(2*n+1/2).
a(n) ~ exp(-1 + 4*sqrt(n) - 2*n) * n^(2*n + 1/2)/2 * (1 + 19/(24*sqrt(n)) + 589/(1152*n)). - Vaclav Kotesovec, Sep 27 2016

A276965 Square row sums of the triangle of Lah numbers (A105278).

Original entry on oeis.org

1, 1, 5, 73, 2017, 86801, 5289301, 430814665, 45052534913, 5868875082817, 930114039075301, 175964489469769001, 39125942325820605025, 10092849114680961297553, 2987365449592984040715317, 1005030253302269078318250601
Offset: 0

Views

Author

Emanuele Munarini, Sep 27 2016

Keywords

Links

Crossrefs

Cf. A000262, A105278, A008297.

Programs

  • Mathematica
    Table[HypergeometricPFQ[{1-n,1-n,-n,-n},{1},1],{n,0,100}]
  • Maxima
    makelist(hypergeometric([-n+1,-n+1,-n,-n],[1],1),n,0,12);
  • PARI
    concat([1], for(n=1,25, print1(sum(k=0,n, binomial(n,k)^2*binomial(n-1,k-1)^2*((n-k)!)^2), ", "))) \\ G. C. Greubel, Jun 05 2017
  • Perl
    use ntheory ":all"; for my $n (0..20) { say "$n ",vecsum(map{my $l=stirling($n,$,3); vecprod($l,$l); } 0..$n) } # _Dana Jacobsen, Mar 16 2017

Formula

a(n) = Sum_{k=0..n} lah(n,k)^2.
a(n) = Sum_{k=0..n} binomial(n,k)^2*binomial(n-1,k-1)^2*((n-k)!)^2.
a(n) = hypergeometric([-n+1,-n+1,-n,-n],[1],1).
a(n) = (n!)^2 * hypergeometric([-n+1,-n+1],[1,2,2],1) for n > 0.
Recurrence: n*(16*n^3 - 96*n^2 + 185*n - 116)*a(n) = 2*(32*n^6 - 272*n^5 + 930*n^4 - 1668*n^3 + 1670*n^2 - 867*n + 164)*a(n-1) - (n-2)*(96*n^7 - 1056*n^6 + 4646*n^5 - 10500*n^4 + 12990*n^3 - 8644*n^2 + 2827*n - 364)*a(n-2) + 2*(n-3)*(n-2)^3*(32*n^6 - 336*n^5 + 1410*n^4 - 2978*n^3 + 3268*n^2 - 1731*n + 353)*a(n-3) - (n-4)^2*(n-3)^3*(n-2)^4*(16*n^3 - 48*n^2 + 41*n - 11)*a(n-4). - Vaclav Kotesovec, Sep 27 2016
a(n) ~ n^(2*n - 3/4) * exp(4*sqrt(n) - 2*n - 1) / (2^(3/2) * sqrt(Pi)) * (1 + 31/(96*sqrt(n)) + 937/(18432*n)). - Vaclav Kotesovec, Sep 27 2016

A076126 Triangle T(n,k) of associated Lah numbers, n>=2, k=1..floor(n/2).

Original entry on oeis.org

2, 6, 24, 12, 120, 120, 720, 1080, 120, 5040, 10080, 2520, 40320, 100800, 40320, 1680, 362880, 1088640, 604800, 60480, 3628800, 12700800, 9072000, 1512000, 30240, 39916800, 159667200, 139708800, 33264000, 1663200, 479001600
Offset: 2

Views

Author

Vladeta Jovovic, Vladimir Baltic, Oct 31 2002

Keywords

Comments

Number of partitions of {1,..,n} into k lists of size >1, where a list means an ordered subset, cf. A008297.

Examples

			2; 6; 24, 12; 120,120; 720,1080,120; 5040,10080, 2520; ...

Crossrefs

Row sums give A052845, A008306, A008299.

Formula

T(n, k) = n!/k!*binomial(n-k-1, k-1), n>=2, k=1..floor(n/2). G.f.: G.f.: Sum_{n>=2, k=1..floor(n/2)} T(n, k)*x^n*y^k/n! = exp(x^2*y/(1-x))-1.

A223513 Triangle T(n,k) represents the coefficients of (x^11*d/dx)^n, where n=1,2,3,...

Original entry on oeis.org

1, 11, 1, 231, 33, 1, 7161, 1287, 66, 1, 293601, 61215, 4125, 110, 1, 14973651, 3476781, 279840, 10065, 165, 1, 913392711, 230534073, 21106701, 924000, 20790, 231, 1, 64850882481, 17511845967, 1771323246, 89482701, 2483250, 38346, 308, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Comments

Generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Examples

			1;
11,1;
231,33,1;
7161,1287,66,1;
293601,61215,4125,110,1;
14973651,3476781,279840,10065,165,1;
913392711,230534073,21106701,924000,20790,23,1;
64850882481,17511845967,1771323246,89482701,2483250,38346,308,1;

Crossrefs

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^11*diff(b[j-1],x$1);
end do;

A223519 Triangle T(n,k) represents the coefficients of (x^17*d/dx)^n, where n=1,2,3,...

Original entry on oeis.org

1, 17, 1, 561, 51, 1, 27489, 3111, 102, 1, 1786785, 232815, 9945, 170, 1, 144729585, 20877615, 1058250, 24225, 255, 1, 14038769745, 2190735855, 125644365, 3480750, 49980, 357, 1, 1586380981185, 263782657215, 16639837830, 529411365, 9328410, 92106, 476, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Comments

Generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Examples

			1;
17,1;
561,51,1;
27489,3111,102,1;
1786785,232815,9945,170,1;
144729585,20877615,1058250,24225,255,1;
14038769745,2190735855,125644365,3480750,49980,357,1;
1586380981185,263782657215,16639837830,529411365,9328410,92106,476,1;

Crossrefs

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^17*diff(b[j-1],x$1);
end do;

A223520 Triangle T(n,k) represents the coefficients of (x^18*d/dx)^n, where n=1,2,3,....

Original entry on oeis.org

1, 18, 1, 630, 54, 1, 32760, 3492, 108, 1, 2260440, 277200, 11160, 180, 1, 194397840, 26376840, 1259280, 27180, 270, 1, 20022977520, 2937589200, 158601240, 4140360, 56070, 378, 1, 2402757302400, 375471270720, 22286940480, 667865520, 11093040, 103320, 504, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Comments

Generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Examples

			1;
18,1;
630,54,1;
32760,3492,108,1;
2260440,277200,11160,180,1;
194397840,26376840,1259280,27180,270,1;
20022977520,2937589200,158601240,4140360,56070,378,1;
2402757302400,375471270720,22286940480,667865520,11093040,103320,504,1

Crossrefs

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^18*diff(b[j-1],x$1);
end do;

A223521 Triangle T(n,k) represents the coefficients of (x^19*d/dx)^n, where n=1,2,3,...

Original entry on oeis.org

1, 19, 1, 703, 57, 1, 38665, 3895, 114, 1, 2822545, 326895, 12445, 190, 1, 256851595, 32896885, 1484280, 30305, 285, 1, 27996823855, 3875508945, 197651965, 4878440, 62510, 399, 1, 3555596629585, 524061968815, 29372612430, 831849165, 13067250, 115178, 532, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Comments

Generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Examples

			1;
19,1;
703,57,1;
38665,3895,114,1;
2822545,326895,12445,190,1;
256851595,32896885,1484280,30305,285,1;
27996823855,3875508945,197651965,4878440,62510,399,1;
3555596629585,524061968815,29372612430,831849165,13067250,115178,532,1;

Crossrefs

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^19*diff(b[j-1],x$1);
end do;
Previous Showing 101-107 of 107 results.

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