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 11-19 of 19 results.

A317021 Expansion of Product_{k>=1} 1/(1 - x^k)^((3*k-1)*binomial(k+2,3)/2).

Original entry on oeis.org

1, 1, 11, 51, 216, 861, 3477, 13367, 50377, 184667, 664484, 2345230, 8142476, 27825576, 93750686, 311682789, 1023547782, 3322634928, 10669887669, 33916213669, 106776876109, 333111724130, 1030264525744, 3160359629535, 9618807643826, 29057370625281, 87153154537437
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 19 2018

Keywords

Comments

Euler transform of A002419.

Links

Crossrefs

Cf. A000391, A002419, A274998, A279219, A305653, A317017, A317019, A317020.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(add(
      (3*d-1)*binomial(d+2, 3)/2*d, d=numtheory
      [divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 19 2018
  • Mathematica
    nmax = 26; CoefficientList[Series[Product[1/(1 - x^k)^((3 k - 1) Binomial[k + 2, 3]/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 26; CoefficientList[Series[Exp[Sum[x^k (1 + 5 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (3 d - 1)/12, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 26}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002419(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 5*x^k)/(k*(1 - x^k)^5)).
a(n) ~ 1/(2^(1987/2160) * 3^(713/1080) * 7^(173/2160) * n^(1253/2160) * Pi^(7/360)) * exp(-1/72 + (1/12-Zeta'(-1))/6 - Zeta(3)/(30 * Pi^2) + (111 * Zeta(5))/(200 * Pi^4) - (7056 * Zeta(3) * Zeta(5)^2)/Pi^12 - (592704 * Zeta(5)^3)/(5 * Pi^14) + (43016085504 * Zeta(5)^5)/(5 * Pi^24) + (2 * Zeta'(-3))/3 + ((-7 * (7/2)^(1/6) * Pi)/(3200 * 3^(2/3)) + (14 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(3) * Zeta(5))/Pi^7 + (1029 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(5)^2)/(5 * Pi^9) - (17978688 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(5)^4)/Pi^19) * n^(1/6) + (-((7/6)^(1/3) * Zeta(3))/(2 * Pi^2) - (7 * 3^(2/3) * (7/2)^(1/3) * Zeta(5))/(5 * Pi^4) + (75264 * 6^(2/3) * 7^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7/2) * Pi)/60 - (1008 * sqrt(14) * Zeta(5)^2)/Pi^9) * sqrt(n) + ((6 * 6^(1/3) * 7^(2/3) * Zeta(5))/Pi^4) * n^(2/3) + ((2 * (2/7)^(1/6) * 3^(2/3) * Pi)/5) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

A322653 Numbers that are sums of consecutive octagonal pyramidal numbers (A002414).

Original entry on oeis.org

0, 1, 9, 10, 30, 39, 40, 70, 100, 109, 110, 135, 205, 231, 235, 244, 245, 364, 366, 436, 466, 475, 476, 540, 595, 730, 765, 800, 830, 839, 840, 904, 1045, 1135, 1270, 1305, 1340, 1370, 1379, 1380, 1386, 1669, 1794, 1810, 1900, 2035, 2105, 2135, 2144, 2145, 2275, 2350, 2431, 2714
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 21 2018

Keywords

Links

Crossrefs

Cf. A002414, A002419, A320728, A321450, A322468, A322479, A322637, A322651, A322652.

A104716 Triangle T(n,k) = (2k-3+4n)*(k-1-n)*(k-2-n)/6, 1<=k<=n.

Original entry on oeis.org

1, 7, 3, 22, 13, 5, 50, 34, 19, 7, 95, 70, 46, 25, 9, 161, 125, 90, 58, 31, 11, 252, 203, 155, 110, 70, 37, 13, 372, 308, 245, 185, 130, 82, 43, 15, 525, 444, 364, 287, 215, 150, 94, 49, 17, 715, 615, 516, 420, 329, 245, 170, 106, 55, 19, 946, 825, 705, 588, 476, 371, 275, 190, 118, 61, 21
Offset: 1

Views

Author

Gary W. Adamson, Mar 20 2005

Keywords

Comments

The triangle is created by the matrix product A158405 * A004736, regarding both as infinite lower triangular matrices, rest of the terms filled in with zeros.
Apparently, row n contains the initial terms of row 2n-2 of A177877. - R. J. Mathar, Aug 31 2011

Examples

			First few rows are:
1;
7, 3;
22, 13, 5;
50, 34, 19, 7;
95, 70, 46, 25, 9;
...

Crossrefs

Cf. A104715, A002419 (row sums).

Programs

  • Maple
    A104716 := proc(n,k) (2*k-3+4*n)*(k-1-n)*(k-2-n)/6 ; end proc:
seq(seq(A104716(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Aug 31 2011

Extensions

Closed-form definition by R. J. Mathar, Aug 31 2011

A136526 Coefficients polynomials B(x, n) = ((1 + a + b)*x - c)*B(x, n-1) - a*b*B(x, n-2) with a = 3, b = 2, and c = 0.

Original entry on oeis.org

1, 0, 1, -6, 0, 6, 0, -42, 0, 36, 36, 0, -288, 0, 216, 0, 468, 0, -1944, 0, 1296, -216, 0, 4536, 0, -12960, 0, 7776, 0, -4104, 0, 38880, 0, -85536, 0, 46656, 1296, 0, -51840, 0, 311040, 0, -559872, 0, 279936, 0, 32400, 0, -544320, 0, 2379456, 0, -3639168, 0, 1679616
Offset: 0

Views

Author

Roger L. Bagula, Mar 23 2008

Keywords

Examples

			Triangle begins as:
     1;
     0,     1;
    -6,     0,      6;
     0,   -42,      0,    36;
    36,     0,   -288,     0,    216;
     0,   468,      0, -1944,      0,   1296;
  -216,     0,   4536,     0, -12960,      0,    7776;
     0, -4104,      0, 38880,      0, -85536,       0, 46656;
  1296,     0, -51840,     0, 311040,      0, -559872,     0, 279936;

References

  • Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 93

Links

Crossrefs

Cf. A000567, A002414, A002419, A016921, A027810, A030192, A034265, A051843, A054487, A055848, A081106, A136531.

Programs

  • Magma
    f:= func< n,k | k eq 0 select (-1)^Floor(n/2) else (-1)^Floor((n-k)/2)*6^Floor((k-1)/2)*(1/k)*(6*Floor((n-k)/2) +k)*Binomial(Floor((n-k)/2) +k-1, k-1) >;
A136526:= func< n,k | ((n+k+1) mod 2)*6^Floor(n/2)*f(n,k) >;
[A136526(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 22 2022
  • Mathematica
    (* First program *)
a= (b+1)/(b-1); c=0; b=2;
B[x_, n_]:= B[x, n]= If[n<2, x^n, ((1+a+b)*x -c)*B[x, n-1] -a*b*B[x, n-2]];
Table[CoefficientList[B[x,n], x], {n,0,10}]//Flatten
(* Second program *)
B[x_, n_]:= 6^(n/2)*(ChebyshevU[n, Sqrt[3/2]*x] -(5*x/Sqrt[6])*ChebyshevU[n-1, Sqrt[3/2]*x]);
Table[CoefficientList[B[x, n], x]/6^Floor[n/2], {n,0,16}]//Flatten (* G. C. Greubel, Sep 22 2022 *)
  • SageMath
    def f(n,k):
    if (k==0): return (-1)^(n//2)
    else: return (-1)^((n-k)//2)*6^((k-1)//2)*(1/k)*(6*((n-k)//2) + k)*binomial(((n-k)//2) +k-1, k-1)
def A136526(n,k): return ((n+k+1)%2)*6^(n//2)*f(n,k)
flatten([[A136526(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 22 2022

Formula

T(n, k) = coefficients of the polynomials defined by B(x, n) = ((1 + a + b)*x - c)*B(x, n - 1) - a*b*B(x, n - 2) with B(x, 0) = 1, B(x, 1) = x, a = 3, b = 2, and c = 0.
From G. C. Greubel, Sep 22 2022: (Start)
T(n, k) = coefficients of the polynomials defined by B(x, n) = 6^(n/2)*(ChebyshevU(n, sqrt(3/2)*x) - (5*x/sqrt(6))*ChebyshevU(n-1, sqrt(3/2)*x)).
T(n, k) = (1/2)*(1+(-1)^(n+k))*6^floor(n/2)*f(n, k), where f(n, k) = (-1)^floor((n -k)/2)*6^floor((k-1)/2)*(1/k)*(6*floor((n-k)/2) + k)*binomial(floor((n-k)/2) + k -1, k-1), for k >= 1, and (-1)^floor(n/2) for k = 0.
T(n, 0) = (1/2)*(1+(-1)^n)*(-6)^floor(n/2).
T(n, 1) = (1/2)*(1-(-1)^n)*(-6)^floor((n-1)/2)*A016921(floor((n-1)/2)), n >= 1.
T(n, 2) = (1/2)*(1+(-1)^n)*(-1)^(1+Floor((n+1)/2))*6^floor((n+1)/2)*A000567(floor( (n+1)/2)), n >= 2.
T(n, 3) = (1/2)*(1-(-1)^n)*(-6)^floor((n+1)/2)*A002414(floor((n-1)/2)), n >= 3.
T(n, 4) = (3/2)*(1+(-1)^n)*(-6)^floor((n+1)/2)*A002419(floor((n-1)/2)), n >= 4.
T(n, 5) = 18*(1-(-1)^n)*(-6)^floor((n-1)/2)*A051843(floor((n-3)/2)), n >= 5.
T(n, n) = 6^(n-1) + (5/6)*[n=0].
T(n, n-2) = -6*A081106(n-2), n >= 2.
Sum_{k=0..n} T(n, k) = -6*A030192(n-3), n>= 0.
Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] - 5*[n=2].
G.f.: (1 - 5*x*y)/(1 - 6*x*y + 6*y^2). (End)

Extensions

Edited by G. C. Greubel, Sep 22 2022

A193607 Augmentation of the triangle A011973. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 20, 7, 1, 16, 80, 131, 37, 1, 25, 220, 806, 1085, 265, 1, 36, 490, 3130, 9360, 10952, 2402, 1, 49, 952, 9325, 48224, 124498, 130852, 26371, 1, 64, 1680, 23317, 183569, 813886, 1876101, 1809430, 340272
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
(col 2): A000290 (the squares)
(col 3)= 2*A002419

Examples

			First five rows of A193607:
1
1...1
1...4....2
1...9....20...7
1...16...80...131...37

Crossrefs

Cf. A011973, A193091, A002419.

Programs

  • Mathematica
    p[n_, k_] := Binomial[2 n - k, k];
Table[p[n, k], {n, 0, 7}, {k, 0, n}]  (* A011973 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 9}]] (* A193607 *)
Flatten[Table[v[n], {n, 0, 8}]]

A125235 Triangle with the partial column sums of the octagonal numbers.

Original entry on oeis.org

1, 8, 1, 21, 9, 1, 40, 30, 10, 1, 65, 70, 40, 11, 1, 96, 135, 110, 51, 12, 1, 133, 231, 245, 161, 63, 13, 1, 176, 364, 476, 406, 224, 76, 14, 1, 225, 540, 840, 882, 630, 300, 90, 15, 1, 280, 765, 1380, 1722, 1512, 930, 390, 105, 16, 1
Offset: 1

Views

Author

Gary W. Adamson, Nov 24 2006

Keywords

Comments

"Partial column sums" means the octagonal numbers are the 1st column, the 2nd column are the partial sums of the 1st column, the 3rd column are the partial sums of the 2nd, etc.
Row sums are 1, 9, 31, 81, 187, 405, 847 = 7*(2^n-1) - 6*n. - R. J. Mathar, Sep 06 2011

Examples

			First few rows of the triangle:
   1;
   8,   1;
  21,   9,   1;
  40,  30,  10,   1;
  65,  70,  40,  11,   1;
  96, 135, 110,  51,  12,   1;
  ...

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover (1966), p. 189.

Crossrefs

Cf. A000567, A002414, A002419, A051843, A027810, A125232 - A125234.

Programs

  • PARI
    t(n, k) = if (n <0, 0, if (k==1, n*(3*n-2), if (k > 1, t(n-1,k-1) + t(n-1,k))));
tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(t(n, k), ", ");); print(););} \\ Michel Marcus, Mar 04 2014

Formula

T(n,1) = A000567(n).
T(n,k) = T(n-1,k-1) + T(n-1,k), k>1.
T(n,2) = A002414(n-1).
T(n,3) = A002419(n-2).
T(n,4) = A051843(n-4).
T(n,5) = A027810(n-6).

Extensions

More terms from Michel Marcus, Mar 04 2014

A287143 Expansion of x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).

Original entry on oeis.org

0, 1, 4, 9, 21, 35, 65, 95, 155, 210, 315, 406, 574, 714, 966, 1170, 1530, 1815, 2310, 2695, 3355, 3861, 4719, 5369, 6461, 7280, 8645, 9660, 11340, 12580, 14620, 16116, 18564, 20349, 23256, 25365, 28785, 31255, 35245, 38115, 42735, 46046, 51359, 55154, 61226, 65550, 72450, 77350, 85150, 90675, 99450, 105651, 115479
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 15 2017

Keywords

Links

Crossrefs

Cf. A002413, A002418, A002419, A002624, A027656, A085787, A212246, A290055.

Programs

  • Mathematica
    CoefficientList[Series[x (1 + 3 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 52}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 4, 9, 21, 35, 65, 95, 155}, 53]

Formula

G.f.: x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
Generalized 4-dimensional figurate numbers (A002418): (5*n - 1)*binomial(n + 2,3)/4, n = 0,+1,-3,+2,-4,+3,-5, ...
Convolution of the sequences A027656 and A085787.
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(5*(2*n^2+10*n+3)-3*(2*n+5)*(-1)^n)/3072. - Luce ETIENNE, Nov 18 2017

A290055 Expansion of x*(1 + 4*x + x^2)/((1 - x)^5*(1 + x)^4).

Original entry on oeis.org

0, 1, 5, 10, 26, 40, 80, 110, 190, 245, 385, 476, 700, 840, 1176, 1380, 1860, 2145, 2805, 3190, 4070, 4576, 5720, 6370, 7826, 8645, 10465, 11480, 13720, 14960, 17680, 19176, 22440, 24225, 28101, 30210, 34770, 37240, 42560, 45430, 51590, 54901, 61985, 65780, 73876, 78200, 87400, 92300, 102700, 108225, 119925, 126126
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 15 2017

Keywords

Comments

More generally, the generalized 4-dimensional figurate numbers are convolution of the sequence {1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...} with generalized polygonal numbers (A195152).

Links

Crossrefs

Cf. A001082, A002414, A002419, A002624, A027656, A195152, A287143.

Programs

  • Mathematica
    CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 51}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 5, 10, 26, 40, 80, 110, 190}, 52]
  • PARI
    x='x+O('x^99); concat(0, Vec(x*(1+4*x+x^2)/((1-x)^5*(1 + x)^4))) \\ Altug Alkan, Aug 15 2017

Formula

G.f.: x*(1 + 4*x + x^2)/((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
Generalized 4-dimensional figurate numbers (A002419): (3*n - 1)*binomial(n + 2,3)/2, n = 0,+1,-3,+2,-4,+3,-5, ...
Convolution of the sequences A027656 and A001082 (with offset 0).
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(6*n^2+30*n+5-(2*n+5)*(-1)^n)/1536. - Luce ETIENNE, Nov 18 2017

A187673 Partial sums of the tricapped prism numbers A005920.

Original entry on oeis.org

1, 10, 43, 125, 290, 581, 1050, 1758, 2775, 4180, 6061, 8515, 11648, 15575, 20420, 26316, 33405, 41838, 51775, 63385, 76846, 92345, 110078, 130250, 153075, 178776, 207585, 239743, 275500, 315115, 358856
Offset: 0

Views

Author

Jonathan Vos Post, Mar 12 2011

Keywords

Links

Crossrefs

Cf. A005920.

Programs

  • Mathematica
    Accumulate[LinearRecurrence[{4,-6,4,-1},{1,9,33,82},40]] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,10,43,125,290},40] (* Harvey P. Dale, Feb 15 2015 *)

Formula

a(n) = Sum_{i=0..n} A005920(i).
a(n) = (n+2)*(n+1)*(9*n^2 + 19*n + 12)/24.
a(n) = A002419(n+1) + A050534(n+1).
G.f.: ( -1-5*x-3*x^2 ) / (x-1)^5. - R. J. Mathar, Mar 29 2011

Extensions

Typo in formula fixed by Colin Barker, Apr 19 2013
Previous Showing 11-19 of 19 results.

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