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 1-10 of 19 results. Next

A213275 Number A(n,k) of words w where each letter of the k-ary alphabet occurs n times and for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 15, 7, 1, 1, 1, 120, 105, 106, 19, 1, 1, 1, 720, 945, 2575, 1075, 56, 1, 1, 1, 5040, 10395, 87595, 115955, 13326, 174, 1, 1, 1, 40320, 135135, 3864040, 19558470, 7364321, 188196, 561, 1, 1
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2012

Keywords

Comments

The words counted by A(n,k) have length n*k.

Examples

			A(0,k) = A(n,0) = 1: the empty word.
A(n,1) = 1: (a)^n for alphabet {a}.
A(1,2) = 2: ab, ba for alphabet {a,b}.
A(1,3) = 6: abc, acb, bac, bca, cab, cba for alphabet {a,b,c}.
A(2,2) = 3: aabb, abab, baab.
A(2,3) = 15: aabbcc, aabcbc, aacbbc, ababcc, abacbc, abcabc, acabbc, acbabc, baabcc, baacbc, bacabc, bcaabc, caabbc, cababc, cbaabc.
A(3,2) = 7: aaabbb, aababb, aabbab, abaabb, ababab, baaabb, baabab.
Square array A(n,k) begins:
  1, 1,   1,      1,         1,             1,                 1, ...
  1, 1,   2,      6,        24,           120,               720, ...
  1, 1,   3,     15,       105,           945,             10395, ...
  1, 1,   7,    106,      2575,         87595,           3864040, ...
  1, 1,  19,   1075,    115955,      19558470,        4622269345, ...
  1, 1,  56,  13326,   7364321,    7236515981,    10915151070941, ...
  1, 1, 174, 188196, 586368681, 3745777177366, 40684710729862072, ...

Links

Crossrefs

Rows n=0-10 give: A000012, A000142, A001147, A213863, A213864, A213865, A213866, A213867, A213868, A213869, A213870.
Columns k=0+1, 2-10 give: A000012, A005807(n-1) for n>0, A213873, A213874, A213875, A213876, A213877, A213878, A213871, A213872.
Main diagonal gives A213862.
Cf. A213276.

Programs

  • Maple
    A:= (n, k)-> b([n$k]):
b:= proc(l) option remember;
      `if`({l[]} minus {0}={}, 1, add(`if`(g(l, i),
       b(subsop(i=l[i]-1, l)), 0), i=1..nops(l)))
    end:
g:= proc(l, i) local j;
      if l[i]<1     then return false
    elif l[i]>1     then for j from i+1 to nops(l) do
      if l[i]<=l[j] then return false
    elif l[j]>0     then break
      fi od fi; true
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    a[n_, k_] := b[Array[n&, k]];
b[l_] := b[l] = If[l ~Complement~ {0} == {}, 1, Sum[If[g[l, i], b[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, Length[l]}]];
g[l_, i_] := Module[{j},
     If[l[[i]] < 1, Return[False],
     If[l[[i]] > 1, For[j = i+1, j <= Length[l], j++,
     If[l[[i]] <= l[[j]], Return[False],
     If[l[[j]] > 0, Break[]]]]]]; True];
Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A257783 Number T(n,k) of words w of length n such that each letter of the k-ary alphabet is used at least once and for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 6, 0, 1, 7, 12, 24, 0, 1, 12, 35, 60, 120, 0, 1, 25, 87, 210, 360, 720, 0, 1, 44, 232, 609, 1470, 2520, 5040, 0, 1, 89, 599, 1961, 4872, 11760, 20160, 40320, 0, 1, 160, 1591, 5952, 17649, 43848, 105840, 181440, 362880, 0, 1, 321, 4202, 19255, 60465, 176490, 438480, 1058400, 1814400, 3628800
Offset: 0

Views

Author

Alois P. Heinz, May 08 2015

Keywords

Comments

Row n is the inverse binomial transform of the n-th row of array A213276.

Examples

			T(5,2) = 12: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, baaaa, baaab, baaba.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  2;
  0, 1,  3,   6;
  0, 1,  7,  12,   24;
  0, 1, 12,  35,   60,  120;
  0, 1, 25,  87,  210,  360,   720;
  0, 1, 44, 232,  609, 1470,  2520,  5040;
  0, 1, 89, 599, 1961, 4872, 11760, 20160, 40320;

Links

Crossrefs

Columns k=0-10 give: A000007, A057427, A321838, A321839, A321840, A321841, A321842, A321843, A321844, A321845, A321846.
Main diagonal gives A000142.
T(n+1,n) = A001710(n+1) (for n>0).
Cf. A213276.

Programs

  • Mathematica
    g[l_, i_] := Module[{j}, If[l[[i]] < 1, Return[False], If[l[[i]] > 1, For[j = i + 1, j <= Length[l], j++, If[l[[i]] <= l[[j]], Return[False], If[l[[j]] > 0, Break[]]]]]]; True];
b[l_] := b[l] = If[Complement[l, {0}] == {}, 1, Sum[If[g[l, i], b[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, Length[l]}]];
h[n_, k_, m_, l_] := h[n, k, m, l] = If[n == 0 && k === 0, b[l], If[k == 0 || n > 0 && n < m, 0, Sum[h[n - j, k - 1, Max[m, j], Join[{j}, l]], {j, Max[1, m], n}] + h[n, k - 1, m, Join[{0}, l]]]];
A[n_, k_] := h[n, k, 0, {}];
T[n_, k_] := Sum[(-1)^i*Binomial[k, i]*A[n, k - i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz in A213276 *)

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A213276(n,k-i).

A213283 Number of 4-length words w over n-ary alphabet such that for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.

Original entry on oeis.org

0, 1, 9, 36, 118, 315, 711, 1414, 2556, 4293, 6805, 10296, 14994, 21151, 29043, 38970, 51256, 66249, 84321, 105868, 131310, 161091, 195679, 235566, 281268, 333325, 392301, 458784, 533386, 616743, 709515, 812386, 926064, 1051281, 1188793, 1339380, 1503846
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2012

Keywords

Examples

			a(0) = 0: no word of length 4 is possible for an empty alphabet.
a(1) = 1: aaaa for alphabet {a}.
a(2) = 9: aaaa, aaab, aaba, aabb, abaa, abab, baaa, baab, bbbb for alphabet {a,b}.
a(3) = 36: aaaa, aaab, aaac, aaba, aabb, aabc, aaca, aacb, aacc, abaa, abab, abac, abca, acaa, acab, acac, acba, baaa, baab, baac, baca, bbbb, bbbc, bbcb, bbcc, bcaa, bcbb, bcbc, caaa, caab, caac, caba, cbaa, cbbb, cbbc, cccc for alphabet {a,b,c}.

Links

Crossrefs

Row n=4 of A213276.

Programs

  • Maple
    a:= n-> n*(-9+(17+(-8+2*n)*n)*n)/2:
seq(a(n), n=0..40);

Formula

a(n) = n*(-9+17*n-8*n^2+2*n^3)/2.
G.f.: x*(1+4*x+x^2+18*x^3)/(1-x)^5.

A213284 Number of 5-length words w over n-ary alphabet such that for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.

Original entry on oeis.org

0, 1, 14, 74, 276, 895, 2506, 6104, 13224, 26061, 47590, 81686, 133244, 208299, 314146, 459460, 654416, 910809, 1242174, 1663906, 2193380, 2850071, 3655674, 4634224, 5812216, 7218725, 8885526, 10847214, 13141324, 15808451, 18892370, 22440156, 26502304
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2012

Keywords

Examples

			a(0) = 0: no word of length 5 is possible for an empty alphabet.
a(1) = 1: aaaaa for alphabet {a}.
a(2) = 14: aaaaa, aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, baaaa, baaab, baaba, bbbbb for alphabet {a,b}.

Links

Crossrefs

Row n=5 of A213276.

Programs

  • Maple
    a:= n-> n*(94+(-204+(155+(-45+6*n)*n)*n)*n)/6:
seq(a(n), n=0..40);

Formula

a(n) = n*(94-204*n+155*n^2-45*n^3+6*n^4)/6.
G.f.: x*(1+8*x+5*x^2+22*x^3+84*x^4)/(1-x)^6.

A213285 Number of 6-length words w over n-ary alphabet such that for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.

Original entry on oeis.org

0, 1, 27, 165, 712, 2535, 8151, 23527, 60600, 140517, 297595, 584001, 1075152, 1875835, 3127047, 5013555, 7772176, 11700777, 17167995, 24623677, 34610040, 47773551, 64877527, 86815455, 114625032, 149502925, 192820251, 246138777, 311227840, 390081987, 484939335
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2012

Keywords

Examples

			a(0) = 0: no word of length 6 is possible for an empty alphabet.
a(1) = 1: aaaaaa for alphabet {a}.
a(2) = 27: aaaaaa, aaaaab, aaaaba, aaaabb, aaabaa, aaabab, aaabba, aaabbb, aabaaa, aabaab, aababa, aababb, aabbaa, aabbab, abaaaa, abaaab, abaaba, abaabb, ababaa, ababab, baaaaa, baaaab, baaaba, baaabb, baabaa, baabab, bbbbbb for alphabet {a,b}.

Links

Crossrefs

Row n=6 of A213276.

Programs

  • Maple
    a:= n-> n*(-332+(757+(-632+(255+(-48+4*n)*n)*n)*n)*n)/4:
seq(a(n), n=0..40);

Formula

a(n) = n*(-332+757*n-632*n^2+255*n^3-48*n^4+4*n^5)/4.
G.f.: x*(1+20*x-3*x^2+89*x^3+106*x^4+507*x^5) / (1-x)^7.

A213286 Number of 7-length words w over n-ary alphabet such that for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.

Original entry on oeis.org

0, 1, 46, 367, 1805, 7280, 25781, 83916, 250062, 676155, 1662160, 3748261, 7839811, 15370082, 28505855, 50400890, 85502316, 139914981, 221828802, 342014155, 514390345, 756672196, 1091099801, 1545256472, 2152979930, 2955371775, 4001910276, 5351671521
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2012

Keywords

Examples

			a(0) = 0: no word of length 7 is possible for an empty alphabet.
a(1) = 1: aaaaaaa for alphabet {a}.
a(2) = 46: aaaaaaa, aaaaaab, aaaaaba, aaaaabb, aaaabaa, aaaabab, aaaabba, aaaabbb, aaabaaa, aaabaab, aaababa, aaababb, aaabbaa, aaabbab, aaabbba, aabaaaa, aabaaab, aabaaba, aabaabb, aababaa, aababab, aababba, aabbaaa, aabbaab, aabbaba, abaaaaa, abaaaab, abaaaba, abaaabb, abaabaa, abaabab, abaabba, ababaaa, ababaab, abababa, baaaaaa, baaaaab, baaaaba, baaaabb, baaabaa, baaabab, baaabba, baabaaa, baabaab, baababa, bbbbbbb for alphabet {a,b}.

Links

Crossrefs

Row n=7 of A213276.

Programs

  • Maple
    a:= n-> n*(11954+ (-29577 +(27640 +(-12831+(3234+(-420+24*n)*n) *n) *n) *n) *n)/24:
seq(a(n), n=0..40);

Formula

a(n) = n*(11954-29577*n+27640*n^2-12831*n^3+3234*n^4-420*n^5+24*n^6)/24.
G.f.: x*(1+38*x+27*x^2+101*x^3+610*x^4+693*x^5+3570*x^6)/(1-x)^8.

A213287 Number of 8-length words w over n-ary alphabet such that for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.

Original entry on oeis.org

0, 1, 91, 869, 4895, 21562, 83728, 296268, 977026, 2990967, 8418649, 21740455, 51758345, 114517208, 237528214, 465636886, 868918932, 1553027197, 2672453415, 4447208761, 7183467523, 11298758534, 17352329324, 26081348272, 38443650358, 55667772435, 79311064261
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2012

Keywords

Examples

			a(0) = 0: no word of length 8 is possible for an empty alphabet.
a(1) = 1: aaaaaaaa for alphabet {a}.
a(2) = 91: aaaaaaaa, aaaaaaab, ..., baababab, bbbbbbbb for alphabet {a,b}.

Links

Crossrefs

Row n=8 of A213276.

Programs

  • Maple
    a:= n-> n*(-417562+ (1092135+ (-1113650+ (587165+ (-175728+ (30520+ (-2880+120*n) *n) *n) *n) *n) *n) *n)/120:
seq(a(n), n=0..40);

Formula

a(n) = n*(-417562 +1092135*n -1113650*n^2 +587165*n^3 -175728*n^4 +30520*n^5 -2880*n^6 +120*n^7)/120.
G.f.: x*(1+82*x +86*x^2 +266*x^3 +1273*x^4 +4234*x^5 +5880*x^6 +28498*x^7) / (1-x)^9.

A213288 Number of 9-length words w over n-ary alphabet such that for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.

Original entry on oeis.org

0, 1, 162, 2074, 13280, 64924, 273248, 1050777, 3754472, 12602451, 39598078, 115470300, 311272072, 777274550, 1808153452, 3946185587, 8137258032, 15957939797, 29935676058, 53988338158, 94013898576, 158665898944, 260355640952, 416527654621, 651260985944
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2012

Keywords

Examples

			a(0) = 0: no word of length 9 is possible for an empty alphabet.
a(1) = 1: aaaaaaaaa for alphabet {a}.
a(2) = 162: aaaaaaaaa, aaaaaaaab, ..., baabababa, bbbbbbbbb for alphabet {a,b}.

Links

Crossrefs

Row n=9 of A213276.

Programs

  • Maple
    a:= n-> n*(3353416+ (-9177198+ (10002755+ (-5796570+ (1984509+ (-416052+ (52920+ (-3780+120*n) *n) *n) *n) *n) *n) *n) *n)/120:
seq(a(n), n=0..40);

Formula

a(n) = n*(3353416 -9177198*n +10002755*n^2 -5796570*n^3 +1984509*n^4 -416052*n^5 +52920*n^6 -3780*n^7 +120*n^8)/120.
G.f.: x*(1+152*x +499*x^2 -290*x^3 +6224*x^4 +6496*x^5 +41203*x^6 +52034*x^7 +256561*x^8) / (1-x)^10.

A213289 Number of 10-length words w over n-ary alphabet such that for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.

Original entry on oeis.org

0, 1, 323, 5168, 37993, 201975, 916966, 3771418, 14486158, 52359693, 178880725, 575581556, 1731294863, 4845394723, 12619979568, 30703918750, 70168864396, 151545355033, 311129635863, 610492421368, 1150383157925, 2090531036111, 3677200683178, 6280769764578
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2012

Keywords

Examples

			a(0) = 0: no word of length 10 is possible for an empty alphabet.
a(1) = 1: aaaaaaaaaa for alphabet {a}.
a(2) = 323: aaaaaaaaaa, aaaaaaaaab, ..., baabababab, bbbbbbbbbb for alphabet {a,b}.

Links

Crossrefs

Row n=10 of A213276.

Programs

  • Maple
    a:= n-> n*(-6044030+ (17209877+ (-19851310+ (12422410+ (-4715472+ (1139127+ (-176832+ (17190+(-960+24*n) *n)*n)*n)*n)*n)*n)*n)*n)/24:
seq(a(n), n=0..40);

Formula

a(n) = n*(-6044030 +17209877*n -19851310*n^2 +12422410*n^3 -4715472*n^4 +1139127*n^5 -176832*n^6 +17190*n^7 -960*n^8 +24*n^9)/24.
G.f.: x*(1+312*x +1670*x^2 -1255*x^3 +15327*x^4 +38264*x^5 +81248*x^6 +406785*x^7 +520730*x^8 +2565718*x^9)/(1-x)^11.

A213290 Number of n-length words w over binary alphabet such that for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.

Original entry on oeis.org

1, 2, 4, 5, 9, 14, 27, 46, 91, 162, 323, 589, 1177, 2179, 4357, 8152, 16303, 30746, 61491, 116689, 233377, 445095, 890189, 1704795, 3409589, 6552379, 13104757, 25258601, 50517201, 97617061, 195234121, 378098956, 756197911, 1467343306, 2934686611, 5704370761
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2012

Keywords

Examples

			a(0) = 1: the empty word.
a(1) = 2: a, b for alphabet {a,b}.
a(2) = 4: aa, ab, ba, bb.
a(3) = 5: aaa, aab, aba, baa, bbb.
a(4) = 9: aaaa, aaab, aaba, aabb, abaa, abab, baaa, baab, bbbb.
a(5) = 14: aaaaa, aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, baaaa, baaab, baaba, bbbbb.

Links

Crossrefs

Column k=2 of A213276.
Cf. A001405, A057427, A182172.

Programs

  • Maple
    b:= n-> `if`(n<0, 0, binomial(n, ceil(n/2))):
a:= n-> b(n) +b(n-2) +`if`(n>0, 1, 0):
seq(a(n), n=0..40);

Formula

a(n) = A001405(n) + A001405(n-2) + A057427(n).
a(n) = A182172(n,2) + A182172(n-2,2) + A057427(n).
Showing 1-10 of 19 results. Next

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