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.

Showing 1-6 of 6 results.

A331436 Array read by antidiagonals: A(n,k) is the number of n element multisets of n element multisets of a k-set.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 21, 20, 1, 0, 1, 5, 55, 220, 70, 1, 0, 1, 6, 120, 1540, 3060, 252, 1, 0, 1, 7, 231, 7770, 73815, 53130, 924, 1, 0, 1, 8, 406, 30856, 1088430, 5461512, 1107568, 3432, 1, 0, 1, 9, 666, 102340, 11009376, 286243776, 581106988, 26978328, 12870, 1, 0
Offset: 0

Views

Author

Andrew Howroyd, Jan 17 2020

Keywords

Examples

			Array begins:
==================================================================
n\k | 0 1   2       3         4            5              6
----+-------------------------------------------------------------
  0 | 1 1   1       1         1            1              1 ...
  1 | 0 1   2       3         4            5              6 ...
  2 | 0 1   6      21        55          120            231 ...
  3 | 0 1  20     220      1540         7770          30856 ...
  4 | 0 1  70    3060     73815      1088430       11009376 ...
  5 | 0 1 252   53130   5461512    286243776     8809549056 ...
  6 | 0 1 924 1107568 581106988 127860662755 13949678575756 ...
    ...
The A(2,2) = 6 multisets are:
   {{1,1}, {1,1}},
   {{1,1}, {1,2}},
   {{1,1}, {2,2}},
   {{1,2}, {1,2}},
   {{1,2}, {2,2}},
   {{2,2}, {2,2}}.

Links

Crossrefs

Rows n=0..3 are A000012, A001477, A002817, A140236.
Columns k=0..10 are A000007, A000012, A000984, A099121, A099122, A099123, A099124, A099125, A099126, A099127, A099128.
Min diagonal is A331477.

Programs

  • PARI
    T(n,k)={binomial(binomial(n + k - 1, n) + n - 1, n)}
{ for(n=0, 7, for(k=0, 7, print1(T(n,k), ", ")); print) }

Formula

A(n,k) = binomial(binomial(n + k - 1, n) + n - 1, n).

A329753 Doubly square pyramidal numbers.

Original entry on oeis.org

0, 1, 55, 1015, 9455, 56980, 255346, 924490, 2850730, 7757035, 19096385, 43312841, 91753025, 183453270, 349074740, 636310340, 1117143236, 1897397285, 3129084635, 5026125195, 7884086595, 12104671656, 18225763270, 26957923950, 39228339150, 56233289775, 79500340101, 110961532605
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

Links

Crossrefs

Cf. A000290, A000330, A000583, A140236, A329754, A329755, A329756, A329757.

Programs

  • Mathematica
    A000330[n_] := n (n + 1) (2 n + 1)/6; a[n_] := A000330[A000330[n]]; Table[a[n], {n, 0, 27}]
Table[Sum[k^2, {k, 0, n (n + 1) (2 n + 1)/6}], {n, 0, 27}]
nmax = 27; CoefficientList[Series[x (1 + 45 x + 510 x^2 + 1660 x^3 + 1715 x^4 + 519 x^5 + 30 x^6)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 55, 1015, 9455, 56980, 255346, 924490, 2850730, 7757035}, 28]

Formula

G.f.: x*(1 + 45*x + 510*x^2 + 1660*x^3 + 1715*x^4 + 519*x^5 + 30*x^6)/(1 - x)^10.
a(n) = A000330(A000330(n)).
a(n) = Sum_{k=0..A000330(n)} A000290(k).
a(n) = n *(2*n+1) *(n+2) *(n+1) *(2*n^2-n+3) *(2*n^3+3*n^2+n+3) /648. - R. J. Mathar, Nov 28 2019

A329754 Doubly pentagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 126, 3078, 32800, 213750, 1008126, 3783976, 11985408, 33297075, 83338750, 191592126, 410450976, 828497488, 1589341950, 2917620000, 5154021376, 8801526501, 14585352318, 23529456550, 37052820000, 57089119626, 86233820926, 127923156648, 186649920000, 268221484375, 380065968126
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

Links

Crossrefs

Cf. A000326, A002411, A140236, A232713, A329753, A329755, A329756, A329757.

Programs

  • Mathematica
    A002411[n_] := n^2 (n + 1)/2; a[n_] := A002411[A002411[n]]; Table[a[n], {n, 0, 26}]
Table[Sum[k (3 k - 1)/2, {k, 0, n^2 (n + 1)/2}], {n, 0, 26}]
nmax = 26; CoefficientList[Series[x (1 + 116 x + 1863 x^2 + 7570 x^3 + 9350 x^4 + 3474 x^5 + 304 x^6 + 2 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 126, 3078, 32800, 213750, 1008126, 3783976, 11985408, 33297075}, 27]

Formula

G.f.: x*(1 + 116*x + 1863*x^2 + 7570*x^3 + 9350*x^4 + 3474*x^5 + 304*x^6 + 2*x^7)/(1 - x)^10.
a(n) = A002411(A002411(n)).
a(n) = Sum_{k=0..A002411(n)} A000326(k).
a(n) = n^4 *(n^3+n^2+2) *(n+1)^2 /16. - R. J. Mathar, Nov 28 2019

A329755 Doubly hexagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 252, 7337, 84575, 576080, 2795121, 10700382, 34388362, 96606475, 243939410, 564840991, 1217275137, 2469392562, 4757404575, 8765621740, 15534503236, 26603512517, 44196596312, 71459197125, 112756874195, 174046844356, 263335062397, 391232840362, 571628456750, 822490729775
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

Links

Crossrefs

Cf. A000384, A002412, A063249, A140236, A329753, A329754, A329756, A329757.

Programs

  • Mathematica
    A002412[n_] := n (n + 1) (4 n - 1)/6; a[n_] := A002412[A002412[n]]; Table[a[n], {n, 0, 25}]
Table[Sum[k (2 k - 1), {k, 0, n (n + 1) (4 n - 1)/6}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[x (1 + 242 x + 4862 x^2 + 22425 x^3 + 30465 x^4 + 12424 x^5 + 1248 x^6 + 13 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 252, 7337, 84575, 576080, 2795121, 10700382, 34388362, 96606475}, 26]

Formula

G.f.: x*(1 + 242*x + 4862*x^2 + 22425*x^3 + 30465*x^4 + 12424*x^5 + 1248*x^6 + 13*x^7)/(1 - x)^10.
a(n) = A002412(A002412(n)).
a(n) = Sum_{k=0..A002412(n)} A000384(k).
a(n) = n *(4*n-1) *(n+1) *(4*n^3+3*n^2-n+6) *(8*n^3+6*n^2-2*n-3) / 648 . - R. J. Mathar, Nov 28 2019

A329756 Doubly heptagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 456, 14976, 181780, 1273970, 6293756, 24395756, 79119496, 223821235, 568280240, 1321714636, 2858876956, 5817509516, 11237224740, 20751835560, 36849296016, 63215722181, 105182448536, 170297734920, 269047574180, 415753060646, 629674964556, 936359517556
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

Links

Crossrefs

Cf. A000566, A002413, A140236, A264891, A329753, A329754, A329755, A329757.

Programs

  • Mathematica
    A002413[n_] := n (n + 1) (5 n - 2)/6; a[n_] := A002413[A002413[n]]; Table[a[n], {n, 0, 25}]
Table[Sum[k (5 k - 3)/2, {k, 0, n (n + 1) (5 n - 2)/6}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[x (1 + 446 x + 10461 x^2 + 52420 x^3 + 75580 x^4 + 32544 x^5 + 3504 x^6 + 44 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 456, 14976, 181780, 1273970, 6293756, 24395756, 79119496, 223821235}, 26]

Formula

G.f.: x*(1 + 446*x + 10461*x^2 + 52420*x^3 + 75580*x^4 + 32544*x^5 + 3504*x^6 + 44*x^7)/(1 - x)^10.
a(n) = A002413(A002413(n)).
a(n) = Sum_{k=0..A002413(n)} A000566(k).
a(n) = n *(5*n-2) *(n+1) *(5*n^3+3*n^2-2*n+6) *(25*n^3+15*n^2-10*n-12)/1296. - R. J. Mathar, Nov 28 2019

A329757 Doubly octagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 765, 27435, 345415, 2469420, 12352956, 48294610, 157609530, 447989355, 1141711615, 2663460261, 5775482505, 11777133550, 22789550070, 42150245460, 74946834916, 128723876325, 214401953745, 347453633935, 549386792955, 849592039296, 1287617552320, 1915941609990, 2803320397950, 4038796372975
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

Links

Crossrefs

Cf. A000567, A002414, A140236, A264892, A329753, A329754, A329755, A329756.

Programs

  • Mathematica
    A002414[n_] := n (n + 1) (2 n - 1)/2; a[n_] := A002414[A002414[n]]; Table[a[n], {n, 0, 25}]
Table[Sum[k (3 k - 2), {k, 0, n (n + 1) (2 n - 1)/2}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[x (1 + 755 x + 19830 x^2 + 105370 x^3 + 158255 x^4 + 70629 x^5 + 7930 x^6 + 110 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 765, 27435, 345415, 2469420, 12352956, 48294610, 157609530, 447989355}, 26]

Formula

G.f.: x*(1 + 755*x + 19830*x^2 + 105370*x^3 + 158255*x^4 + 70629*x^5 + 7930*x^6 + 110*x^7)/(1 - x)^10.
a(n) = A002414(A002414(n)).
a(n) = Sum_{k=0..A002414(n)} A000567(k).
a(n) = n *(2*n-1) *(n+1) *(2*n^3+n^2-n+2) *(2*n^3+n^2-n-1) /8 . - R. J. Mathar, Nov 28 2019
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.