A000538 -id:A000538 - cp's OEIS frontend

A218115 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5 * y^k] * x^n/n ) = Sum_{n>=0,k=0..n} T(n,k)x^ny^k, as a triangle of coefficients T(n,k) read by rows.

1, 1, 1, 1, 17, 1, 1, 98, 98, 1, 1, 354, 2251, 354, 1, 1, 979, 23803, 23803, 979, 1, 1, 2275, 158367, 617036, 158367, 2275, 1, 1, 4676, 773842, 8763293, 8763293, 773842, 4676, 1, 1, 8772, 3031668, 82498785, 241082026, 82498785, 3031668, 8772, 1, 1, 15333

Offset: 0

Paul D. Hanna, Oct 21 2012

Compare g.f. to that of the following triangle variants:
* Pascal's: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)*y^k] * x^n/n );
* Narayana: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*y^k] * x^n/n );
* A181143: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*y^k] * x^n/n );
* A181144: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*y^k] * x^n/n );
* A218116: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^6*y^k] * x^n/n ).

			G.f.: A(x,y) = 1 + (1+y)*x + (1+17*y+y^2)*x^2 + (1+98*y+98*y^2+y^3)*x^3 + (1+354*y+2251*y^2+354*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^5*y + y^2)*x^2/2
+ (1 + 3^5*y + 3^5*y^2 + y^3)*x^3/3
+ (1 + 4^5*y + 6^5*y^2 + 4^5*y^3 + y^4)*x^4/4
+ (1 + 5^5*y + 10^5*y^2 + 10^5*y^3 + 5^5*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 17, 1;
1, 98, 98, 1;
1, 354, 2251, 354, 1;
1, 979, 23803, 23803, 979, 1;
1, 2275, 158367, 617036, 158367, 2275, 1;
1, 4676, 773842, 8763293, 8763293, 773842, 4676, 1;
1, 8772, 3031668, 82498785, 241082026, 82498785, 3031668, 8772, 1;
1, 15333, 10057620, 575963523, 4066874561, 4066874561, 575963523, 10057620, 15333, 1; ...
Note that column 1 forms the sum of fourth powers (A000538).

Cf. A000538 (column 1), A218117 (row sums).

Cf. variants: A001263 (Narayana), A181143, A181144, A218116.

{T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^5*y^j)*x^m/m)+O(x^(n+1))), n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A359320 Maximal coefficient of (1 + x) * (1 + x^16) * (1 + x^81) * ... * (1 + x^(n^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 9, 13, 17, 24, 34, 53, 84, 130, 177, 290, 500, 797, 1300, 2066, 3591, 6090, 10298, 17330, 29888, 50811, 88358, 153369, 280208, 481289, 845090, 1474535, 2703811, 4808816, 8329214, 14806743, 27529781, 48859783, 87674040, 156471632

Offset: 0

Ilya Gutkovskiy, Dec 25 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A000538, A000583, A025591, A160235, A298859, A359319.

f:= proc(n) local i; max(coeffs(expand(mul(1+x^(i^4), i=1..n)))) end proc:
map(f, [$1..50]); # Robert Israel, Dec 26 2022

a(n) = vecmax(Vec(prod(k=1, n, 1+x^(k^4)))); \\ Michel Marcus, Dec 26 2022

from collections import Counter
def A359320(n):
    c = {0:1,1:1}
    for i in range(2,n+1):
        j, d = i**4, Counter(c)
        for k in c:
            d[k+j] += c[k]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

a(38)-a(50) from Seiichi Manyama, Dec 26 2022

A004538 a(n) = 3n^2 + 3n - 1.

Original entry on oeis.org

-1, 5, 17, 35, 59, 89, 125, 167, 215, 269, 329, 395, 467, 545, 629, 719, 815, 917, 1025, 1139, 1259, 1385, 1517, 1655, 1799, 1949, 2105, 2267, 2435, 2609, 2789, 2975, 3167, 3365, 3569, 3779, 3995, 4217, 4445

Offset: 0

N. J. A. Sloane, Eric T. Lane (ERICLANE(AT)UTCVM.UTC.EDU)

Numbers k such that (4*k + 7)/3 is a square. - Bruno Berselli, Sep 11 2018

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First differences of A033445.

Cf. A028896, A003215, A077588, A000538, A000330.

[3*n^2 + 3*n -1: n in [0..50]]; // G. C. Greubel, Sep 10 2018

Table[5*Sum[k^4,{k,1,n}]/Sum[k^2,{k,1,n}], {n,1,20}] (* Alexander Adamchuk, Apr 12 2006 *)
Table[3n^2+3n-1,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{-1,5,17},40] (* Harvey P. Dale, Jan 18 2019 *)

a(n)=3*n^2+3*n-1 \\ Charles R Greathouse IV, Jun 17 2017

From Alexander Adamchuk, Apr 12 2006: (Start)
a(n) = 5 * Sum_{k=1..n} k^4 / Sum_{k=1..n} k^2, n > 0.
a(n) = 5 * A000538(n) / A000330(n), n > 0. (End)
a(n) = a(n-1) + 6*n with a(0)=-1. - Vincenzo Librandi, Nov 18 2010
From G. C. Greubel, Sep 10 2018: (Start)
G.f.: (-1 + 8*x - x^2)/(1 - x)^3.
E.g.f.: (-1 + 6*x + 3*x^2)*exp(x). (End)
Sum_{n>=0} 1/a(n) = ( psi(1/2+sqrt(21)/6) - psi(1/2-sqrt(21)/6)) /sqrt(21) = -0.6286929... R. J. Mathar, Apr 24 2024

A060452 Let v = (1,4,9,...,n^2), x = (0,1,2,4,6,...) [first n terms of A002620]; a(n) = v.v * x.x - (v.x)^2.

Original entry on oeis.org

0, 1, 6, 38, 107, 350, 728, 1752, 3090, 6215, 9878, 17654, 26117, 42924, 60256, 93024, 125460, 184509, 241110, 341110, 434511, 595562, 742808, 991640, 1215110, 1586403, 1914822, 2452646, 2922185, 3681560, 4337024, 5385600, 6281704, 7701561, 8904294, 10793862, 12381939, 14858038, 16924440, 20124440, 22778042, 26862143, 30229430, 35383062, 39609933, 46046276, 51299936, 59262560, 65733500, 75499125

Offset: 1

N. J. A. Sloane and Vinay Vaishampayan, Apr 09 2001

nonn

N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, Fat Struts: Constructions and a Bound, Proceedings Information Theory Workshop, Taormino, Italy, 2009. [Cached copy]
N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, A Note on Projecting the Cubic Lattice, Discrete and Computational Geometry, Vol. 46 (No. 3, 2011), 472-478.
N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, The Lifting Construction: A General Solution to the Fat Strut Problem, Proceedings International Symposium on Information Theory (ISIT), 2010, IEEE Press. [Cached copy]
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

Cf. A002620, A000538, A059859. Agrees with A060453 for first 37 terms.

fv := n->1/30*n*(1+n)*(2*n+1)*(3*n^2+3*n-1); # this is A000538
f1 := n->1/160*(n-1)*(1+n)*(2*n^3+5*n^2+2*n-5);
f2 := n->1/160*n*(n+2)*(2*n^3+n^2-2*n+4);
f7 := n->if n mod 2 = 0 then f2(n) else f1(n) end if; # this is A059859
f3 := n->1/20*n^5+1/8*n^4+1/24*n^3-11/120*n-1/8*n^2;
f4 := n->1/20*n^5+1/8*n^4+1/24*n^3+1/30*n;
f5:-n-> if `mod`(n,2) = 0 then f4(n) else f3(n) end if; # this is A060453
A060452 := n->f7(n)*fv(n)-f5(n)^2;

Table[Module[{nn=n,v,x},v=Range[nn]^2;x=Floor[v/4];v.v x.x-(v.x)^2],{n,50}] (* or *) LinearRecurrence[{1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{0,1,6,38,107,350,728,1752,3090,6215,9878,17654,26117},50] (* Harvey P. Dale, Aug 10 2021 *)

G.f. -x^2*(1+5*x+26*x^2+39*x^3+66*x^4+39*x^5+26*x^6+5*x^7+x^8) / ( (1+x)^6*(x-1)^7 ). - R. J. Mathar, Apr 04 2012

A133821 Triangle whose rows are sequences of increasing fourth powers: 1; 1,16; 1,16,81; ... .

Original entry on oeis.org

1, 1, 16, 1, 16, 81, 1, 16, 81, 256, 1, 16, 81, 256, 625, 1, 16, 81, 256, 625, 1296, 1, 16, 81, 256, 625, 1296, 2401, 1, 16, 81, 256, 625, 1296, 2401, 4096, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000

Offset: 1

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,16,1,16,81,1,16,81,256,..., analogous to A002260.

			Triangle starts
1;
1, 16;
1, 16; 81;
1, 16, 81, 256;
1, 16, 81, 256, 625;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A000538 (row sums), A002260, A133819, A133820, A133824.

a133821 n k = a133821_tabl !! (n-1) !! (k-1)
a133821_row n = a133821_tabl !! (n-1)
a133821_tabl = map (`take` (tail a000583_list)) [1..]
-- Reinhard Zumkeller, Nov 11 2012

Module[{nn=10,fp},fp=Range[(nn(nn+1))/2]^4;Table[TakeList[fp,{n}],{n,nn}]]//Flatten (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 29 2020 *)

O.g.f.: (1+11qx+11q^2x^2+q^3x^3)/((1-x)(1-qx)^5) = 1 + x(1 + 16q) + x^2(1 + 16q + 81q^2) + ... . Cf. 4th row of A008292.

Offset changed by Reinhard Zumkeller, Nov 11 2012

A140144 a(1)=1, a(n)=a(n-1)+n^1 if n odd, a(n)=a(n-1)+ n^0 if n is even.

Original entry on oeis.org

1, 2, 5, 6, 11, 12, 19, 20, 29, 30, 41, 42, 55, 56, 71, 72, 89, 90, 109, 110, 131, 132, 155, 156, 181, 182, 209, 210, 239, 240, 271, 272, 305, 306, 341, 342, 379, 380, 419, 420, 461, 462, 505, 506, 551, 552, 599, 600, 649, 650, 701, 702, 755, 756, 811, 812, 869

Offset: 1

Artur Jasinski, May 12 2008

Equals triangle A177990 * [1,2,3,...]. - Gary W. Adamson, May 16 2010

Girtrude Hamm, Classification of lattice triangles by their two smallest widths, arXiv:2304.03007 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.
Cf. A177990. - Gary W. Adamson, May 16 2010
Cf. A002378 (even bisection), A028387 (odd bisection).

a = {}; r = 1; s = 0; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a

From R. J. Mathar, Feb 22 2009: (Start)
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(-1-x-x^2+x^3)/ ((1+x)^2*(x-1)^3). (End)
a(n) = Sum_{k=1..n} k^(k mod 2). - Wesley Ivan Hurt, Nov 20 2021

A145217 a(n) is the self-convolution series of the sum of 4th powers of the first n natural numbers.

Original entry on oeis.org

1, 32, 418, 3104, 16003, 64064, 213060, 614976, 1587333, 3742816, 8190182, 16832608, 32795399, 61021312, 109078664, 188234880, 314856201, 512202912, 812698666, 1260762272, 1916300683, 2858972864, 4193345740, 6055075520

Offset: 1

Abdullahi Umar, Oct 05 2008

			a(3) = 418 because 1(3^4)+(2^4)(2^4)+(3^4)1= 418

A. Umar, B. Yushau and B. M. Ghandi, (2006), "Patterns in convolution of two series", in Stewart, S. M., Olearski, J. E. and Thompson, D. (Eds), Proceedings of the Second Annual Conference for Middle East Teachers of Science, Mathematics and Computing (pp. 95-101). METSMaC: Abu Dhabi.
A. Umar, B. Yushau and B. M. Ghandi, "Convolution of two series", Australian Senior Maths. Journal, 21(2) (2007), 6-11.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 2. - _N. J. A. Sloane_, Mar 23 2014
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

a(n) = Conv(A000538, A000538).

[Binomial(n+2,3)*(n*(n+2)*(n^4+4*n^3+8*n^2+8*n+6)+24)/105: n in [1..40]]; // Vincenzo Librandi, Mar 24 2014

f:=n->(n^9+20*n^3-21*n)/630;
[seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 23 2014

CoefficientList[Series[(1 + x)^2 (1 + 10 x + x^2)^2/(1 - x)^10, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,32,418,3104,16003,64064,213060,614976,1587333,3742816},30] (* Harvey P. Dale, May 19 2021 *)

a(n) = C(n+2,3)*(n*(n+2)*(n^4+4*n^3+8*n^2+8*n+6)+24)/105.

G.f.: x*(1+x)^2*(1+10*x+x^2)^2/(1-x)^10. [Colin Barker, May 25 2012]

A164307 Primes in A081175.

Original entry on oeis.org

3, 5, 17, 257, 65537

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

The 6th term is too large to include in the data section (see Example section or the b-file).
Primes of the form sum_{j=1..u} j^x for some x>0, u>1. (Since the case of x=1 leads to the triangular numbers with no additional primes, this is equivalent to the definition.)
Primes in A000330 (x=2), or in A000537 (x=3), or in A000538 (x=4), or in A000539 (x=5) etc. See A164312 for the corresponding x values.

			a(1) = 1^1 + 2^1 = 3.
a(2) = 1^2 + 2^2 = 5.
a(3) = 1^4 + 2^4 = 17.
a(4) = 1^8 + 2^8 = 257.
a(5) = 1^16 + 2^16 = 65537.
a(6) = 1^1440 + 2^1440 + 3^1440 + 4^1440 + 5^1440 = 3.287049497374559048967261852*10^1006 = 3287049497374559048967261852 ... 458593539025033893379.

N. J. A. Sloane, Table of n, a(n) for n = 1..6

Cf. A000215, A070592, A019434, A092506, A093179, A100270, A123599.

lst={};Do[s=0;Do[If[PrimeQ[s+=n^x],AppendTo[lst,s];Print[Date[],s]],{n, 4!}],{x,7!}];lst

Edited by R. J. Mathar, Aug 22 2009

Corrected by N. J. A. Sloane, Nov 23 2015 at the suggestion of Jaroslav Krizek.

A302352 a(n) = Sum_{k=0..n} k^4binomial(2n-k,n).

Original entry on oeis.org

0, 1, 19, 155, 936, 4884, 23465, 107107, 472600, 2036838, 8631206, 36119798, 149724940, 616104450, 2520629685, 10265200035, 41650094640, 168481778790, 679847488650, 2737640364810, 11005139655744, 44176226269728, 177114113623194, 709364594864910, 2838599638596176, 11350436081373340

Offset: 0

Ilya Gutkovskiy, Apr 06 2018

nonn

Main diagonal of iterated partial sums array of fourth powers (starting with the first partial sums). For nonnegative integers see A002054, for squares see A265612, for cubes see A293550.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000538, A000583, A002054, A101089, A101090, A101091, A265612, A293550, A302353.

Table[Sum[k^4 Binomial[2 n - k, n], {k, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[x (1 + 11 x + 11 x^2 + x^3)/(1 - x)^(n + 6), {x, 0, n}], {n, 0, 25}]
Table[2^(2 n + 1) n (75 n^3 + 52 n^2 - 3 n - 4) Gamma[n + 3/2]/(Sqrt[Pi] Gamma[n + 6]), {n, 0, 25}]
CoefficientList[Series[(24 - 180 x + 410 x^2 - 285 x^3 + 31 x^4 + Sqrt[1 - 4 x] (-24 + 132 x - 194 x^2 + 65 x^3 - x^4))/(2 Sqrt[1 - 4 x] x^5), {x, 0, 25}], x]
CoefficientList[Series[E^(2 x) (-576 + 360 x - 244 x^2 + 75 x^3) BesselI[0, 2 x]/x^3 + E^(2 x) (576 - 360 x + 532 x^2 - 255 x^3 + 75 x^4) BesselI[1, 2 x]/x^4, {x, 0, 25}], x]* Range[0, 25]!

a(n) = sum(k=0, n, k^4*binomial(2*n-k,n)); \\ Michel Marcus, Apr 07 2018

a(n) = [x^n] x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^(n+6).
a(n) = 2^(2*n+1)*n*(75*n^3 + 52*n^2 - 3*n - 4)*Gamma(n+3/2)/(sqrt(Pi)*Gamma(n+6)).
a(n) ~ 75*2^(2*n+1)/sqrt(Pi*n).

A081175 Numbers of the form Sum_{i=1..k} i^j, j >= 1, k >= 1.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 14, 15, 17, 21, 28, 30, 33, 36, 45, 55, 65, 66, 78, 91, 98, 100, 105, 120, 129, 136, 140, 153, 171, 190, 204, 210, 225, 231, 253, 257, 276, 285, 300, 325, 351, 354, 378, 385, 406, 435, 441, 465, 496, 506, 513, 528, 561, 595, 630, 650, 666, 703

Offset: 1

N. J. A. Sloane, Apr 18 2003

Union of sums of k-th powers, for k >= 1.

			30 is in the set because 30 = 1^2 + 2^2 + 3^2 + 4^2 (j=2, k=4).

Robert Israel, Table of n, a(n) for n = 1..10000
Michael Penn, an excruciatingly deep dive into the power sum., YouTube video, 2022.

Cf. A000217, A000330, A000537, A000538, A000539, A000540, A000541, A000542, A007487, A023002.

For primes in this sequence see A164307.

N:= 1000: # to get all terms <= N
A:=select(`<=`,{1, seq(seq(sum(i^k,i=1..m), m=2..floor((N*(k+1))^(1/(k+1)))),k = 1 ..ilog2(N-1))},N):
sort(convert(A,list)); # Robert Israel, Jan 26 2015

Take[ Union[ Flatten[ Table[ Sum[ i^j, {i, 1, n}], {j, 1, 9}, {n, 1, 40}]]], 60]

Corrected and extended by Robert G. Wilson v, May 08 2003

cp's OEIS Frontend

A218115 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5 * y^k] * x^n/n ) = Sum_{n>=0,k=0..n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows.

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A359320 Maximal coefficient of (1 + x) * (1 + x^16) * (1 + x^81) * ... * (1 + x^(n^4)).

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A004538 a(n) = 3*n^2 + 3*n - 1.

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A060452 Let v = (1,4,9,...,n^2), x = (0,1,2,4,6,...) [first n terms of A002620]; a(n) = v.v * x.x - (v.x)^2.

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A133821 Triangle whose rows are sequences of increasing fourth powers: 1; 1,16; 1,16,81; ... .

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A140144 a(1)=1, a(n)=a(n-1)+n^1 if n odd, a(n)=a(n-1)+ n^0 if n is even.

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A145217 a(n) is the self-convolution series of the sum of 4th powers of the first n natural numbers.

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A164307 Primes in A081175.

A218115 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5 * y^k] * x^n/n ) = Sum_{n>=0,k=0..n} T(n,k)x^ny^k, as a triangle of coefficients T(n,k) read by rows.

A004538 a(n) = 3n^2 + 3n - 1.

A302352 a(n) = Sum_{k=0..n} k^4binomial(2n-k,n).