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.

User: Adi Dani

Adi Dani's wiki page.

Adi Dani has authored 23 sequences. Here are the ten most recent ones:

A194197 Number of partitions of 60n into parts <= 6.

Original entry on oeis.org

1, 19858, 436140, 2897747, 11402579, 33377536, 80758518, 171070425, 328507157, 585011614, 981355696, 1568220303, 2407275335, 3572259692, 5150061274, 7241796981, 9963892713, 13449163370, 17847892852, 23328914059, 30080688891, 38312388248, 48254972030, 60162269137
Offset: 0

Views

Author

Adi Dani, Aug 21 2011

Keywords

Comments

Number of partitions of 60n+k, 0<=k<60 into parts <=6 is a polynomial of degree 5 by variable n.

Links

Crossrefs

Cf. A001402.

Programs

  • Mathematica
    Table[1 + (167n + 2325n^2 + 15400n^3 + 47250n^4 + 54000n^5)/6, {n, 0, 25}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,19858,436140,2897747,11402579,33377536},30] (* Harvey P. Dale, Aug 12 2018 *)

Formula

a(n) = 1 +(167*n +2325*n^2 +15400*n^3 +47250*n^4 +54000*n^5)/6.
a(n) = A001402(60*n).
G.f.: (3331*x^5+161052*x^4+578757*x^3+317007*x^2+19852*x+1)/(x-1)^6. [Colin Barker, Jan 31 2013]

A192012 Number of ways to use the elements of set {1,..,k}, 0<=k<=3*n, once each to form a collection of n (possibly empty) sets, each with at most 3 elements.

Original entry on oeis.org

1, 4, 35, 877, 46173, 4108044, 550917287, 103674052788, 26046619272535, 8420151470990221, 3404266960229749907, 1682802564587905472500, 998472258682783813839141, 700281698972322460184258208, 573086115189070229131370358179, 541208343386984031504989621465925
Offset: 0

Views

Author

Adi Dani, Jun 22 2011

Keywords

Comments

Number of partitions of the set {0,1,...,3*n} into n parts of size <=3.

Examples

			a(0) = 1 = card({[e]}) where e denotes the empty set.
a(1) = 4 = card({[e],[1],[12],[123]}).
a(2) = 35 = card({ [e,e],[e,1],[e,12],[1,2],[e,123],[1,23],[2,13],[3,12],
[1,234],[2,134],[3,124],[4,123],[12,34],[13,24],[14,23],[12,345],[13,245],[14,235],[15,324],[23,145],[24,135],[25,134],[34,125],[35,124],[45,123],
[123,456],[124,356],[125,346],[126,345],[134,256],[135,246],[136,245],[145,236],[146,235],[156,234] }).

Links

Crossrefs

Partial sums of A144416.

Programs

  • Mathematica
    Table[Sum[k!/(i!3^(i - j)2^(k + j - 2i))Binomial[i, j] Binomial[j,k + 2j - 3i], {k, 0, 3n}, {i, 0, n}, {j, 0, 3i - k}], {n, 0, 15}]

Formula

a(n) = Sum_{k=0..3*n} Sum_{i=0..n} Sum_{j=0..3*i-k} k! *C(i,j) *C(j,k+2*j-3*i) / (i! * 3^(i-j) * 2^(k+j-2*i) ).

A192396 Square array T(n, k) = floor(((k+1)^n - (1+(-1)^k)/2)/2) read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 4, 4, 2, 0, 0, 8, 13, 8, 2, 0, 0, 16, 40, 32, 12, 3, 0, 0, 32, 121, 128, 62, 18, 3, 0, 0, 64, 364, 512, 312, 108, 24, 4, 0, 0, 128, 1093, 2048, 1562, 648, 171, 32, 4, 0, 0, 256, 3280, 8192, 7812, 3888, 1200, 256, 40, 5, 0
Offset: 0

Views

Author

Adi Dani, Jun 29 2011

Keywords

Comments

T(n,k) is the number of compositions of odd natural numbers into n parts <=k.

Examples

			T(2,4)=12: there are 12 compositions of odd natural numbers into 2 parts <=4
  1: (0,1), (1,0);
  3: (1,2), (2,1), (0,3), (3,0);
  5: (1,4), (4,1), (2,3), (3,2);
  7: (3,4), (4,3).
The table starts
    0,  0,   0,   0,    0,    0, ... A000004;
    0,  1,   1,   2,    2,    3, ... A004526;
    0,  2,   4,   8,   12,   18, ... A007590;
    0,  4,  13,  32,   62,  108, ... A036487;
    0,  8,  40, 128,  312,  648, ... A191903;
    0, 16, 121, 512, 1562, 3888, ... A191902;
    .        .      .       .    ...
with columns: A000004, A000079, A003462, A004171, A128531, A081341, ... .
Antidiagonal triangle begins:
  0;
  0,  0;
  0,  1,   0;
  0,  2,   1,   0;
  0,  4,   4,   2,   0;
  0,  8,  13,   8,   2,   0;
  0, 16,  40,  32,  12,   3,  0;
  0, 32, 121, 128,  62,  18,  3,  0;
  0, 64, 364, 512, 312, 108, 24,  4,  0;

Links

Crossrefs

Rows: A000004, A004526, A007590, A036487, A191902, A191902.
Columns: A000004, A000079, A003462, A004171, A081341, A128531.

Programs

  • Magma
    A192396:= func< n,k | Floor(((k+1)^n - (1+(-1)^k)/2)/2) >;
[A192396(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 11 2023
  • Maple
    A192396 := proc(n,k) (k+1)^n-(1+(-1)^k)/2 ; floor(%/2) ; end proc:
seq(seq( A192396(d-k,k),k=0..d),d=0..10) ; # R. J. Mathar, Jun 30 2011
  • Mathematica
    T[n_, k_]:= Floor[((k+1)^n - (1+(-1)^k)/2)/2];
Table[T[n-k,k], {n,0,12}, {k,0,n}]//Flatten
  • SageMath
    def A192396(n,k): return ((k+1)^n - ((k+1)%2))//2
flatten([[A192396(n-k,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 11 2023

A191900 Number of compositions of odd natural numbers into 7 parts <=n.

Original entry on oeis.org

0, 64, 1093, 8192, 39062, 139968, 411771, 1048576, 2391484, 5000000, 9743585, 17915904, 31374258, 52706752, 85429687, 134217728, 205169336, 306110016, 446935869, 640000000, 900544270, 1247178944, 1702412723, 2293235712, 3051757812, 4015905088
Offset: 0

Views

Author

Adi Dani, Jun 19 2011

Keywords

Examples

			a(1)=64: the 64 compositions of odd numbers into 7 parts <=1
  1:(0,0,0,0,0,0,1)-->7!/(6!1!)= 7
  3:(0,0,0,0,1,1,1)-->7!/(4!3!)=35
  5:(0,0,1,1,1,1,1)-->7!/(2!5!)=21
  7:(1,1,1,1,1,1,1)-->7!/(0!7!)= 1.

Links

Programs

  • Mathematica
    Table[Floor[1/2*((n + 1)^7 - (1 + (-1)^n)/2)], {n, 0, 25}]
  • PARI
    a(n)=((n+1)^7-(1+(-1)^n)/2)/2 \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(n) = ((n + 1)^7 - (1 + (-1)^n)/2)/2.
a(2*n+1) = A191494(2*n+1); a(2*n) = A191494(2*n)-1.
G.f.: x*(64+645*x+1821*x^2+1786*x^3+666*x^4+57*x^5+x^6) / ( (1+x)*(x-1)^8 ). - R. J. Mathar, Jun 22 2011

A191901 Number of compositions of odd natural numbers into 6 parts <= n.

Original entry on oeis.org

0, 32, 364, 2048, 7812, 23328, 58824, 131072, 265720, 500000, 885780, 1492992, 2413404, 3764768, 5695312, 8388608, 12068784, 17006112, 23522940, 32000000, 42883060, 56689952, 74017944, 95551488, 122070312, 154457888, 193710244, 240945152, 297411660, 364500000, 443751840
Offset: 0

Views

Author

Adi Dani, Jun 19 2011

Keywords

Examples

			a(1)=32 compositions of odd numbers into 6 parts <=1.
1:(0,0,0,0,0,1)-->6!/(5!1!)= 6
3:(0,0,0,1,1,1)-->6!/(3!3!)=20
5:(0,1,1,1,1,1)-->6!/(1!5!)= 6
-------------------------------------
                            32

Links

Programs

  • Magma
    [((n + 1)^6 - (1 + (-1)^n)/2)/2 : n in [0..25]]; // Vincenzo Librandi, Jul 03 2011
  • Maple
    A191901:=n->((n + 1)^6 - (1 + (-1)^n)/2)/2: seq(A191901(n), n=0..50); # Wesley Ivan Hurt, Apr 10 2017
  • Mathematica
    Table[Floor[1/2*((n + 1)^6 - (1 + (-1)^n)/2)], {n, 0, 30}]
LinearRecurrence[{6,-14,14,0,-14,14,-6,1},{0,32,364,2048,7812,23328,58824,131072},40] (* Harvey P. Dale, Apr 12 2015 *)

Formula

a(n) = ((n + 1)^6 - (1 + (-1)^n)/2)/2.
G.f. -4*x*(8+43*x+78*x^2+43*x^3+8*x^4) / ( (1+x)*(x-1)^7 ). - R. J. Mathar, Jun 26 2011
a(0)=0, a(1)=32, a(2)=364, a(3)=2048, a(4)=7812, a(5)=23328, a(6)=58824, a(7)=131072, a(n)=6*a(n-1)-14*a(n-2)+14*a(n-3)-14*a(n-5)+ 14*a(n-6)- 6*a(n-7)+a(n-8). - Harvey P. Dale, Apr 12 2015

A191899 Number of compositions of odd natural numbers into 8 parts <=n.

Original entry on oeis.org

0, 128, 3280, 32768, 195312, 839808, 2882400, 8388608, 21523360, 50000000, 107179440, 214990848, 407865360, 737894528, 1281445312, 2147483648, 3487878720, 5509980288, 8491781520, 12800000000, 18911429680, 27437936768, 39155492640, 55037657088, 76293945312, 104413532288
Offset: 0

Views

Author

Adi Dani, Jun 19 2011

Keywords

Examples

			a(1)=128 compositions of odd numbers into 8 parts <=1
1:(0,0,0,0,0,0,0,1)-->8!/(7!1!)= 8
3:(0,0,0,0,0,1,1,1)-->8!/(5!3!)=56
5:(0,0,0,1,1,1,1,1)-->8!/(3!5!)=56
7:(0,1,1,1,1,1,1,1)-->8!/(1!7!)= 8
-------------------------------------
                               128

Links

Programs

  • Mathematica
    Table[Floor[1/2*((n + 1)^8 - (1 + (-1)^n)/2)], {n, 0, 25}]
LinearRecurrence[{8,-27,48,-42,0,42,-48,27,-8,1},{0,128,3280,32768,195312,839808,2882400,8388608,21523360,50000000},30] (* Harvey P. Dale, Aug 30 2016 *)
  • PARI
    a(n)=1/2*((n+1)^8-(1+(-1)^n)/2) \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(n) = 1/2*((n + 1)^8 - (1 + (-1)^n)/2).
G.f.: -16*x*(8*x^6+141*x^5+624*x^4+974*x^3+624*x^2+141*x+8) / ((x-1)^9*(x+1)). - Colin Barker, May 16 2013

A191902 Number of compositions of odd positive integers into 5 parts <= n.

Original entry on oeis.org

0, 16, 121, 512, 1562, 3888, 8403, 16384, 29524, 50000, 80525, 124416, 185646, 268912, 379687, 524288, 709928, 944784, 1238049, 1600000, 2042050, 2576816, 3218171, 3981312, 4882812, 5940688, 7174453, 8605184, 10255574, 12150000, 14314575
Offset: 0

Views

Author

Adi Dani, Jun 19 2011

Keywords

Examples

			a(1)=16: the 16 compositions of odd numbers into 5 parts <= 1 are
1: (0,0,0,0,1) --> 5!/(4!1!) =  5;
3: (0,0,1,1,1) --> 5!/(2!3!) = 10;
5: (1,1,1,1,1) --> 5!/(0!5!) =  1.

Links

Crossrefs

Cf. A191484.

Programs

  • Magma
    [((n + 1)^5 - (1 + (-1)^n)/2)/2: n in [0..50]]; // Vincenzo Librandi, Jul 04 2011
  • Mathematica
    Table[Floor[1/2*((n + 1)^5 - (1 + (-1)^n)/2)], {n, 0, 30}]
  • PARI
    a(n)=((n+1)^5-(1+(-1)^n)/2)/2 \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(n) = ((n + 1)^5 - (1 + (-1)^n)/2)/2.
From R. J. Mathar, Jun 22 2011: (Start)
a(2n+1) = A191484(2n+1); a(2n) = A191484(2n) - 1.
G.f.: x*(16 + 41*x + 51*x^2 + 11*x^3 + x^4) / ( (1+x)*(x-1)^6 ). (End)

A191903 Number of compositions of odd natural numbers into 4 parts <= n.

Original entry on oeis.org

0, 8, 40, 128, 312, 648, 1200, 2048, 3280, 5000, 7320, 10368, 14280, 19208, 25312, 32768, 41760, 52488, 65160, 80000, 97240, 117128, 139920, 165888, 195312, 228488, 265720, 307328, 353640, 405000, 461760, 524288, 592960, 668168, 750312
Offset: 0

Views

Author

Adi Dani, Jun 19 2011

Keywords

Examples

			a(1) = 8 compositions of odd numbers into 4 parts < 1.
1:(0,0,0,1),(0,0,1,1),(0,1,0,0),(1,0,0,0)
3:(0,1,1,1),(1,0,1,1),(1,1,0,1),(1,1,1,0)

Links

Crossrefs

Cf. A011863, A171714.

Programs

  • Magma
    [((n + 1)^4 - (1 + (-1)^n)/2)/2: n in [0..50]]; // Vincenzo Librandi, Jul 04 2011
  • Mathematica
    Table[Floor[1/2*((n + 1)^4 - (1 + (-1)^n)/2)], {n, 0, 30}]

Formula

a(n) = ((n + 1)^4 - (1 + (-1)^n)/2)/2.
From R. J. Mathar, Jun 22 2011: (Start)
G.f.: 8*x*(1+x+x^2) / ( (1+x)*(1-x)^5 ).
a(n) = 8*A011863(n+1). (End)
a(n) = floor((n+1)^4/2). - Wesley Ivan Hurt, Jun 14 2013
Sum_{n>=1} 1/a(n) = 3/4 + Pi^4/720 - tanh(Pi/2)*Pi/4. - Amiram Eldar, Aug 13 2022

A191745 a(n) = 12*n^3 + 9*n^2 + 2*n.

Original entry on oeis.org

0, 23, 136, 411, 920, 1735, 2928, 4571, 6736, 9495, 12920, 17083, 22056, 27911, 34720, 42555, 51488, 61591, 72936, 85595, 99640, 115143, 132176, 150811, 171120, 193175, 217048, 242811, 270536, 300295, 332160, 366203, 402496, 441111, 482120, 525595, 571608
Offset: 0

Views

Author

Adi Dani, Jun 14 2011

Keywords

Comments

Number of partitions of 12*n+2 into 4 parts.

Examples

			a(1)=23: there are 23 partitions of 12*1+2=14 into 4 parts: [1,1,1,11], [1,1,2,10], [1,1,3,9], [1,1,4,8], [1,1,5,7], [1,1,6,6], [1,2,2,9], [1,2,3,8], [1,2,4,7], [1,2,5,6], [1,3,3,7], [1,3,4,6], [1,3,5,5], [1,4,4,5], [2,2,2,8], [2,2,3,7], [2,2,4,6], [2,2,5,5], [2,3,3,6], [2,3,4,5], [2,4,4,4], [3,3,3,5], [3,3,4,4].

Links

Programs

  • Magma
    [12*n^3+9*n^2+2*n: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011
  • Mathematica
    Table[12n^3 + 9n^2 + 2n, {n, 0, 30}]
LinearRecurrence[{4,-6,4,-1},{0,23,136,411},40] (* Harvey P. Dale, Nov 05 2019 *)
  • PARI
    a(n)=((12*n+9)*n+2)*n /* Charles R Greathouse IV, Jun 14 2011 */

Formula

From Elmo R. Oliveira, Aug 28 2025: (Start)
G.f.: x*(23 + 44*x + 5*x^2)/(x-1)^4.
E.g.f.: x*(23 + 45*x + 12*x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A190705 a(n) = 6*n^2*(2*n + 1).

Original entry on oeis.org

0, 18, 120, 378, 864, 1650, 2808, 4410, 6528, 9234, 12600, 16698, 21600, 27378, 34104, 41850, 50688, 60690, 71928, 84474, 98400, 113778, 130680, 149178, 169344, 191250, 214968, 240570, 268128, 297714, 329400
Offset: 0

Views

Author

Adi Dani, Jun 14 2011

Keywords

Comments

Number of partitions of 12*n + 1 into 4 parts.

Examples

			a(1)=18: there are 18 partitions of 12*1+1=13 into 4 parts:
  [1,1,1,10], [1,1,2,9], [1,1,3,8], [1,1,4,7], [1,1,5,6],
  [1,2,2,8],  [1,2,3,7], [1,2,4,6], [1,2,5,5], [1,3,3,6],
  [1,3,4,5],  [1,4,4,4], [2,2,2,7], [2,2,3,6], [2,2,4,5],
  [2,3,3,5],  [2,3,4,4], [3,3,3,4].

Links

Programs

Formula

a(n) = 6 * A099721(n).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=18, a(2)=120, a(3)=378. - Harvey P. Dale, Mar 20 2016

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