A064043 -id:A064043 - cp's OEIS frontend

A084990 a(n) = n(n^2+3n-1)/3.

0, 1, 6, 17, 36, 65, 106, 161, 232, 321, 430, 561, 716, 897, 1106, 1345, 1616, 1921, 2262, 2641, 3060, 3521, 4026, 4577, 5176, 5825, 6526, 7281, 8092, 8961, 9890, 10881, 11936, 13057, 14246, 15505, 16836, 18241, 19722, 21281, 22920, 24641, 26446, 28337, 30316

Offset: 0

Gary W. Adamson, Jul 16 2003

Row sums of triangle A131782 starting (1, 6, 17, 36, 65, 106, ...). - Gary W. Adamson, Jul 14 2007
a(n) is the number of triples (x,y,z) in {1,2,...,n}^3 with x <= y <= z or x >= y >= z. - Jack Kennedy, Mar 14 2009
a(2*n) is the difference between numbers of nonnegative multiples of 2*n+1 with even and odd digit sum in base 2*n in interval [0, 16*n^4). - Vladimir Shevelev, May 18 2012

			Let n=2. Consider nonnegative multiples of 5 up to 16*2^4 - 1 = 255. There are 52 such numbers and from them only 8 (namely, 35, 50, 55, 115, 140, 200, 205, 220) have an odd digit sum in base 4. Therefore, a(4) = (52 - 8) - 8 = 36. - _Vladimir Shevelev_, May 18 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-queens problem III. Partial queens, arXiv:1402.4886 [math.CO], 2014-2018. See Conjecture 4.4.
Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000290, A000292, A028387, A064043, A077415, A131782.

I:=[0,1,6,17]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Mar 28 2014

A084990:=n->n*(n^2+3*n-1)/3; seq(A084990(k),k=0..100); # Wesley Ivan Hurt, Oct 19 2013

Table[n*(n^2+3*n-1)/3,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2010 *)
CoefficientList[Series[(x + 2 x^2 - x^3)/((1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 28 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,1,6,17},50] (* Harvey P. Dale, Aug 18 2015 *)

a(n) = n*(n^2+3*n-1)/3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 2*A000292(n-1) - 1 (notice offset=-1 in A000292!).
a(n) = (n-1)*(n+1)*(n+3)/3 + 1. - Reinhard Zumkeller, Aug 20 2007
a(n) = A077415(n+1) + 1 for n > 0; a(n) = A000290(n) + A007290(n); a(n+1) = Sum_{k=0..n} A028387(k). - Reinhard Zumkeller, Aug 20 2007
a(2*n) = Sum_{i=0..16*n^4, i==0 (mod 2*n+1)} (-1)^s_(2*n)(i), where s_k(n) is the digit sum of n in base k. - Vladimir Shevelev, May 18 2012
a(2*n) = (2/(2*n+1))*Sum_{i=1..n} tan^4(Pi*i/(2*n+1)). - Vladimir Shevelev, May 23 2012
a(n) = Sum_{i=1..n} i*(i+1)-1. - Wesley Ivan Hurt, Oct 19 2013
G.f.: x*(1+2*x-x^2)/(1-x)^4. - Vincenzo Librandi, Mar 28 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Vincenzo Librandi, Mar 28 2014
a(n) = A064043(n)/3. - Alois P. Heinz, Jul 21 2017
E.g.f.: x*exp(x)*(x^2 + 6*x + 3)/3. - Stefano Spezia, Mar 06 2024

A265080 Array read by antidiagonals, arising from study of remixing keys in public-key cryptography.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 12, 18, 4, 0, 0, 5, 20, 51, 44, 5, 0, 0, 6, 30, 108, 192, 110, 6, 0, 0, 7, 42, 195, 544, 675, 252, 7, 0, 0, 8, 56, 318, 1220, 2540, 2358, 588, 8, 0, 0, 9, 72, 483, 2364, 7145, 11544, 8043, 1304, 9, 0

Offset: 0

N. J. A. Sloane, Jan 01 2016

See Brown (2015) for precise definition.

If you randomly throw n balls into k boxes then T(n,k)/k^n is the expected number of balls in the fullest box. - Henry Bottomley, Mar 20 2021

			Array begins:
  0, 0,   0,   0,    0,    0, ...
  0, 1,   2,   3,    4,    5, ...
  0, 2,   6,  12,   20,   30, ...
  0, 3,  18,  51,  108,  195, ...
  0, 4,  44, 192,  544, 1220, ...
  0, 5, 110, 675, 2540, 7145, ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Daniel R. L. Brown, Bounds on surmising remixed keys, IACR, Report 2015/375, 2015-2016. See Table 1.

Rows n=1..5 are A001477, A002378, A064043, A265081, A265082.
Columns k=1..5 are A001477, A230137, A265083, A265084, A265085.
Main diagonal is A208250.

Q(p)={my(S=Set(p));prod(i=1, #S, (#select(t->t==S[i],p))!)}
T(n,k)={my(s=0); if(n, forpart(p=n, s+=p[#p]*n!*(#p)!*binomial(k,#p) / (prod(i=1,#p,p[i]!) * Q(Vec(p))))); s} \\ Andrew Howroyd, Mar 20 2021

T(n,k) = {n!*polcoef(sum(j=0, n, exp(x + O(x*x^n))^k - sum(i=0, j, x^i/i!, O(x*x^n))^k), n)} \\ Andrew Howroyd, Aug 09 2025

T(n,k) = n! * [x^n] Sum_{j>=0} (exp(x)^k - (Sum_{i=0..j} x^i/i!)^k). - Andrew Howroyd, Aug 09 2025

More terms from Henry Bottomley, Mar 20 2021

A064044 Square array read by antidiagonals of number of length k walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 6, 3, 1, 0, 6, 18, 12, 4, 1, 0, 10, 60, 51, 20, 5, 1, 0, 20, 200, 234, 108, 30, 6, 1, 0, 35, 700, 1110, 624, 195, 42, 7, 1, 0, 70, 2450, 5460, 3760, 1350, 318, 56, 8, 1, 0, 126, 8820, 27405, 23480, 9770, 2556, 483, 72, 9, 1, 0, 252

Offset: 0

Henry Bottomley, Aug 23 2001

E.g.f. of row n equals ( besseli(0,2*y) + y*besseli(1,2*y) )^n. - Paul D. Hanna, Apr 07 2005

			Rows start:
1, 0,  0,   0,    0,     0,      0, ...
1, 1,  2,   3,    6,    10,     20, ...
1, 2,  6,  18,   60,   200,    700, ...
1, 3, 12,  51,  234,  1110,   5460, ...
1, 4, 20, 108,  624,  3760,  23480, ...
1, 5, 30, 195, 1350,  9770,  73300, ...
1, 6, 42, 318, 2556, 21480, 187140, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

Rows include A000007, A001405, A005566, A064036. Columns include A000012, A001477, A002378, A064043. Cf. A064045.

a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
       add(binomial(k, j)*binomial(j, floor(j/2))
       *a(n-1, k-j), j=0..k))
    end:
seq(seq(a(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 06 2014

a[n_, k_] := a[n, k] = If[n == 0, If[k == 0, 1, 0], Sum[Binomial[k, j]*Binomial[j, Floor[j/2]]*a[n-1, k-j], {j, 0, k}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); k!*polcoeff(polcoeff(1/(1-X*besseli(0,2*Y)-X*Y*besseli(1,2*Y)),n,x),k,y)} /* Hanna */

a(n,k) = Sum{j=0..k} C(k,j) B(j) a(n-1,k-j) where B(j) = C(j,[j/2]) = A001405(j) with a(0,0) = 1 and a(0,k) = 0 for k>0.

E.g.f: 1/(1 - x*besseli(0, 2*y) - x*y*besseli(1, 2*y)). - Paul D. Hanna, Apr 07 2005

A185878 Accumulation array of A185877, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 11, 10, 3, 24, 28, 18, 4, 45, 60, 51, 28, 5, 76, 110, 108, 80, 40, 6, 119, 182, 195, 168, 115, 54, 7, 176, 280, 318, 300, 240, 156, 70, 8, 249, 408, 483, 484, 425, 324, 203, 88, 9, 340, 570, 696, 728, 680, 570, 420, 256, 108, 10, 451, 770, 963, 1040, 1015, 906, 735, 528, 315, 130, 11, 584, 1012, 1290, 1428, 1440, 1344, 1162, 920, 648, 380, 154, 12

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ...

See A144112 for the definition of accumulation array.

			Northwest corner:
  1,  4, 11,  24,  45, ...
  2, 10, 28,  60, 110, ...
  3, 18, 51, 108, 195, ...
  4, 28, 80, 168, 300, ...
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185877, A185879.
Row 1 to 3: A006527, A006331, A064043.
Column 1 to 5: A000027, A028552, A140677, 12*A000096, 5*A130861.

f[n_, k_] := k*n*(2*k^2 - 3*k + 3*k*n - 3*n + 7)/6; Table[f[n - k + 1, k], {n,10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(n,k) = k*n*(2*k^2 -3*k +3*k*n -3*n +7)/6, k>=1, n>=1.

cp's OEIS Frontend

A084990 a(n) = n*(n^2+3*n-1)/3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A265080 Array read by antidiagonals, arising from study of remixing keys in public-key cryptography.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A064044 Square array read by antidiagonals of number of length k walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A185878 Accumulation array of A185877, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A084990 a(n) = n(n^2+3n-1)/3.