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.

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A153272 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 4, read by rows.

Original entry on oeis.org

7, 7, 56, 7, 63, 693, 7, 70, 910, 14560, 7, 77, 1155, 21945, 504735, 7, 84, 1428, 31416, 848232, 27143424, 7, 91, 1729, 43225, 1339975, 49579075, 2131900225, 7, 98, 2058, 57624, 2016840, 84707280, 4150656720, 232436776320, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425, 33760034483175
Offset: 0

Views

Author

Roger L. Bagula, Dec 22 2008

Keywords

Comments

Row sums are {7, 63, 763, 15547, 527919, 28024591, 2182864327, 236674216947, 34243215666247, 6391699984166119, 1497639790982770659, ...}.

Examples

			Triangle begins as:
  7;
  7, 56;
  7, 63,  693;
  7, 70,  910, 14560;
  7, 77, 1155, 21945,  504735;
  7, 84, 1428, 31416,  848232, 27143424;
  7, 91, 1729, 43225, 1339975, 49579075, 2131900225;

Links

Crossrefs

Cf. A153270 (m=2), A153271 (m=3), this sequence (m=4).
Cf. A001730, A051579, A051604.

Programs

  • Magma
    m:=4;
function T(n,k)
  if k eq 0 then return NthPrime(m);
  else return (&*[j*n + NthPrime(m): j in [0..k]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 03 2019
  • Maple
    m:=4; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # G. C. Greubel, Dec 03 2019
  • Mathematica
    T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
Table[T[n,k,4], {n,0,10}, {k,0,n}]//Flatten
  • PARI
    T(n,k) = my(m=4); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ G. C. Greubel, Dec 03 2019
  • Sage
    def T(n, k):
    m=4
    if (k==0): return nth_prime(m)
    else: return product(j*n + nth_prime(m) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019

Formula

T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 4.

Extensions

Edited by G. C. Greubel, Dec 03 2019

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

Original entry on oeis.org

1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683
Offset: 1

Views

Author

N. J. A. Sloane, Nov 20 2010

Keywords

Examples

			G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.

Links

Crossrefs

Cf. A000079 S(1,1,-1), A000108 S(0,0,1), A000142 S(1,-1,0), A000244 S(2,1,-2), A000351 S(4,1,-4), A000400 S(5,1,-5), A000420 S(6,1,-6), A000698 S(2,-3,1), A001710 S(1,1,0), A001715 S(1,2,0), A001720 S(1,3,0), A001725 S(1,4,0), A001730 S(1,5,0), A003319 S(1,-2,1), A005411 S(2,-4,1), A005412 S(2,-2,1), A006012 S(-1,2,2), A006318 S(0,1,1), A047891 S(0,2,1), A049388 S(1,6,0), A051604 S(3,1,0), A051605 S(3,2,0), A051606 S(3,3,0), A051607 S(3,4,0), A051608 S(3,5,0), A051609 S(3,6,0), A051617 S(4,1,0), A051618 S(4,2,0), A051619 S(4,3,0), A051620 S(4,4,0), A051621 S(4,5,0), A051622 S(4,6,0), A051687 S(5,1,0), A051688 S(5,2,0), A051689 S(5,3,0), A051690 S(5,4,0), A051691 S(5,5,0), A053100 S(6,1,0), A053101 S(6,2,0), A053102 S(6,3,0), A053103 S(6,4,0), A053104 S(7,1,0), A053105 S(7,2,0), A053106 S(7,3,0), A062980 S(6,-8,1), A082298 S(0,3,1), A082301 S(0,4,1), A082302 S(0,5,1), A082305 S(0,6,1), A082366 S(0,7,1), A082367 S(0,8,1), A105523 S(0,-2,1), A107716 S(3,-4,1), A111529 S(1,-3,2), A111530 S(1,-4,3), A111531 S(1,-5,4), A111532 S(1,-6,5), A111533 S(1,-7,6), A111546 S(1,0,1), A111556 S(1,1,1), A143749 S(0,10,1), A146559 S(1,1,-2), A167872 S(2,-3,2), A172450 S(2,0,-1), A172485 S(-1,-2,3), A177354 S(1,2,1), A292186 S(4,-6,1), A292187 S(3, -5, 1).

Programs

  • Mathematica
    a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)
  • PARI
    {a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */
  • PARI
    S(v1, v2, v3, N=16) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
};
S(6,-4,-1)
\\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
\\ Gheorghe Coserea, May 12 2017

Formula

a(n) = (6*n - 4) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: x / (1 - 7*x / (1 - 5*x / (1 - 13*x / (1 - 11*x / (1 - 19*x / (1 - 17*x / ... )))))). - Michael Somos, Jan 03 2013
a(n) = 3/(2*Pi^2)*int((4*x)^((3*n-1)/2)/(Ai'(x)^2+Bi'(x)^2), x=0..inf), where Ai'(x), Bi'(x) are the derivatives of the Airy functions. [Vladimir Reshetnikov, Sep 24 2013]
a(n) ~ 6^n * (n-1)! / (2*Pi) [Martin + Kearney, 2011, p.16]. - Vaclav Kotesovec, Jan 19 2015
6*x^2*y' = y^2 - (2*x-1)*y - x, where y(x) = Sum_{n>=1} a(n)*x^n. - Gheorghe Coserea, May 12 2017
G.f.: x/(1 - 2*x - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... Cf. A062980. - Peter Bala, May 21 2017

A051546 Third unsigned column of triangle A051339.

Original entry on oeis.org

0, 0, 1, 24, 431, 7155, 117454, 1961470, 33775244, 603682596, 11235811536, 218055250512, 4413843664416, 93156324734304, 2048591287486080, 46898664421553280, 1116592842912341760, 27618683992928743680
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

From Johannes W. Meijer, Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=3,n=7) ~ exp(-x)/x^3*(1 - 24/x + 431/x^2 - 7155/x^3 + 117454/x^4 + ...) leads to the sequence given above. See A163931 and A163932 for more information.
(End)

References

  • Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051339.

Crossrefs

Cf. A001730 (m=0), A051545 (m=1) unsigned columns.

Formula

a(n) = A051339(n, 2)*(-1)^n; e.g.f.: (log(1-x))^2/(2*(1-x)^7).
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = |f(n,2,7)|, for n>=1. - Milan Janjic, Dec 21 2008

A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 24, 12, 4, 6, 1, 120, 60, 20, 30, 5, 10, 1, 720, 360, 120, 180, 30, 60, 6, 90, 15, 20, 1, 5040, 2520, 840, 1260, 210, 420, 42, 630, 105, 140, 7, 210, 21, 35, 1, 40320, 20160, 6720, 10080, 1680, 3360, 336, 5040, 840, 1120, 56
Offset: 0

Views

Author

Tilman Piesk, Nov 04 2014

Keywords

Comments

This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.
Row lengths: A000041 (partition numbers), Row sums: A005651
Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388
Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012
The rightmost columns form the triangle A173333:
n 0 1 2 4 6 10 14 21 (A000041(1,2,3...)-1)
m
1 1
2 2 1
3 6 3 1
4 24 12 4 1
5 120 60 20 5 1
6 720 360 120 30 6 1
7 5040 2520 840 210 42 7 1
8 40320 20160 6720 1680 336 56 8 1
A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

Examples

			Triangle begins:
  n     0    1    2    3   4   5  6   7   8   9 10
m
0       1
1       1
2       2    1
3       6    3    1
4      24   12    4    6   1
5     120   60   20   30   5  10  1
6     720  360  120  180  30  60  6  90  15  20  1

Links

Crossrefs

Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.

A373168 Triangle read by rows: the exponential almost-Riordan array ( exp(x/(1-x)) | 1/(1-x), x ).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 13, 2, 2, 1, 73, 6, 6, 3, 1, 501, 24, 24, 12, 4, 1, 4051, 120, 120, 60, 20, 5, 1, 37633, 720, 720, 360, 120, 30, 6, 1, 394353, 5040, 5040, 2520, 840, 210, 42, 7, 1, 4596553, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 58941091, 362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1
Offset: 0

Views

Author

Stefano Spezia, May 26 2024

Keywords

Examples

			The triangle begins:
     1;
     1,   1;
     3,   1,   1;
    13,   2,   2,  1;
    73,   6,   6,  3,  1;
   501,  24,  24, 12,  4, 1;
  4051, 120, 120, 60, 20, 5, 1;
  ...

Links

Crossrefs

Cf. A000142, A000262 (k=0), A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431.
Triangle A094587 with 1st column A000262.

Programs

  • Mathematica
    T[n_,0]:=n!SeriesCoefficient[Exp[x/(1-x)],{x,0,n}]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[1/(1-x)*x^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

Formula

T(n,0) = n! * [x^n] exp(x/(1-x)); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] 1/(1-x)*x^(k-1).
T(n,3) = A001710(n-1) for n > 2.
T(n,4) = A001715(n-1) for n > 3.
T(n,5) = A001720(n-1) for n > 4.
T(n,6) = A001725(n-1) for n > 5.
T(n,7) = A001730(n-1) for n > 6.
T(n,8) = A049388(n-8) for n > 7.
T(n,9) = A049389(n-9) for n > 8.
T(n,10) = A049398(n-10) for n > 9.
T(n,11) = A051431(n-11) for n > 10.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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