A261781 -id:A261781 - cp's OEIS frontend

A293582 Number of compositions of n where each part i is marked with a word of length i over a quinary alphabet whose letters appear in alphabetical order and all five letters occur at least once in the composition.

541, 13060, 195020, 2327960, 24418640, 235804122, 2152586500, 18883155160, 160908360260, 1341800118020, 11007289244964, 89168468504160, 715330888641680, 5694960569676240, 45067846839572000, 354959016901129928, 2785141532606257120, 21787375678321712160

Offset: 5

Alois P. Heinz, Oct 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=5 of A261781.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(5):
seq(a(n), n=5..30);

b[n_, k_] := b[n, k] = If[n == 0, 1,
    Sum[b[n - j, k] Binomial[j + k - 1, k - 1], {j, 1, n}]];
a[n_] := With[{k = 5}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[5, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

a(n) = 30*a(n-1) - 380*a(n-2) + 2690*a(n-3) - 11944*a(n-4) + 35618*a(n-5) - 74912*a(n-6) + 115104*a(n-7) - 132120*a(n-8) + 114500*a(n-9) - 74888*a(n - 10) + 36504*a(n - 11) - 12888*a(n - 12) + 3120*a(n - 13) - 464*a(n - 14) + 32*a(n - 15). - Vaclav Kotesovec, Oct 14 2017

a(n) ~ c * d^n, where d = 7.72502395887257562679242875427350515911685429396536... is the real root of the equation -2 + 10*d - 20*d^2 + 20*d^3 - 10*d^4 + d^5 = 0 and c = 0.67250239588725756267924287542735051591168542939653... is the real root of the equation -1 - 50*c - 1000*c^2 - 10000*c^3 - 50000*c^4 + 100000*c^5 = 0. - Vaclav Kotesovec, Oct 15 2017

A293583 Number of compositions of n where each part i is marked with a word of length i over a senary alphabet whose letters appear in alphabetical order and all six letters occur at least once in the composition.

Original entry on oeis.org

4683, 155928, 3116220, 48697048, 657516672, 8065687344, 92540869002, 1011476639976, 10662168594984, 109327852591208, 1097238662684028, 10827944900524680, 105430826499237004, 1015590292306277376, 9698300806656595584, 91961212434214073824, 866974686508851897168

Offset: 6

Alois P. Heinz, Oct 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A261781.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(6):
seq(a(n), n=6..30);

a(n) = 42*a(n-1) - 770*a(n-2) + 8190*a(n-3) - 56854*a(n-4) + 275758*a(n-5) - 980010*a(n-6) + 2645668*a(n-7) - 5576808*a(n-8) + 9366788*a(n-9) - 12715312*a(n - 10) + 14078260*a(n - 11) - 12772248*a(n - 12) + 9499064*a(n - 13) - 5769584*a(n - 14) + 2837496*a(n - 15) - 1113568*a(n - 16) + 340784*a(n - 17) - 78416*a(n - 18) + 12768*a(n - 19) - 1312*a(n - 20) + 64*a(n - 21). - Vaclav Kotesovec, Oct 14 2017

A293584 Number of compositions of n where each part i is marked with a word of length i over a septenary alphabet whose letters appear in alphabetical order and all seven letters occur at least once in the composition.

Original entry on oeis.org

47293, 2075948, 53476920, 1058754564, 17866313444, 270907452704, 3807403790792, 50592275219138, 644225577441572, 7936529529027736, 95254972055989564, 1119634204276346052, 12939870424457764200, 147501747088827091436, 1662420626477581539972

Offset: 7

Alois P. Heinz, Oct 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..975

Column k=7 of A261781.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(7):
seq(a(n), n=7..30);

a(n) = 56*a(n-1) - 1400*a(n-2) + 20804*a(n-3) - 206864*a(n-4) + 1472576*a(n-5) - 7857468*a(n-6) + 32533654*a(n-7) - 107414264*a(n-8) + 288967984*a(n-9) - 644267912*a(n - 10) + 1206205784*a(n - 11) - 1915352424*a(n - 12) + 2598569764*a(n - 13) - 3027512680*a(n - 14) + 3038439672*a(n - 15) - 2630187744*a(n - 16) + 1962871608*a(n - 17) - 1260043528*a(n - 18) + 692851920*a(n - 19) - 324225312*a(n - 20) + 127932656*a(n - 21) - 42016752*a(n - 22) + 11279872*a(n - 23) - 2411968*a(n - 24) + 395168*a(n - 25) - 46592*a(n - 26) + 3520*a(n - 27) - 128*a(n - 28). - Vaclav Kotesovec, Oct 14 2017

A293585 Number of compositions of n where each part i is marked with a word of length i over an octonary alphabet whose letters appear in alphabetical order and all eight letters occur at least once in the composition.

Original entry on oeis.org

545835, 30532384, 984910128, 24082101504, 496274574936, 9104663637024, 153620123190816, 2434519831873920, 36763980389367378, 534505149483841568, 7538836344403305280, 103747819539055788640, 1399283448432865901624, 18560930972118370361856

Offset: 8

Alois P. Heinz, Oct 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..925

Column k=8 of A261781.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(8):
seq(a(n), n=8..30);

a(n) = 72*a(n-1) - 2352*a(n-2) + 46452*a(n-3) - 624540*a(n-4) + 6112176*a(n-5) - 45535444*a(n-6) + 266958678*a(n-7) - 1264269754*a(n-8) + 4940034192*a(n-9) - 16203617768*a(n - 10) + 45247660712*a(n - 11) - 108803821608*a(n - 12) + 227386203188*a(n - 13) - 416072786528*a(n - 14) + 670510739364*a(n - 15) - 955987452656*a(n - 16) + 1210032902216*a(n - 17) - 1363000899064*a(n - 18) + 1368396317120*a(n - 19) - 1225293972272*a(n - 20) + 978413136792*a(n - 21) - 696053259552*a(n - 22) + 440365848816*a(n - 23) - 247084003008*a(n - 24) + 122486489680*a(n - 25) - 53377923152*a(n - 26) + 20315614688*a(n - 27) - 6696256832*a(n - 28) + 1890687584*a(n - 29) - 450764768*a(n - 30) + 89006080*a(n - 31) - 14167936*a(n - 32) + 1747264*a(n - 33) - 156672*a(n - 34) + 9088*a(n - 35) - 256*a(n - 36). - Vaclav Kotesovec, Oct 14 2017

A293586 Number of compositions of n where each part i is marked with a word of length i over a nonary alphabet whose letters appear in alphabetical order and all nine letters occur at least once in the composition.

Original entry on oeis.org

7087261, 492006708, 19423259316, 574637640288, 14193955791576, 309660911167464, 6171397007611848, 114853532449557600, 2026594842428425320, 34277110454602760762, 560261324259420037164, 8904738970375872782112, 138290600270036591006520, 2106511986693346884064584

Offset: 9

Alois P. Heinz, Oct 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..885

Column k=9 of A261781.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(9):
seq(a(n), n=9..30);

a(n) = 90*a(n-1) - 3720*a(n-2) + 94140*a(n-3) - 1642368*a(n-4) + 21111972*a(n-5) - 208936444*a(n-6) + 1643906838*a(n-7) - 10544413816*a(n-8) + 56273496182*a(n-9) - 254124223400*a(n - 10) + 984813733064*a(n - 11) - 3313818868728*a(n - 12) + 9777617820932*a(n - 13) - 25505157099056*a(n - 14) + 59222241227144*a(n - 15) - 123105458091224*a(n - 16) + 230174411303404*a(n - 17) - 388610578141384*a(n - 18) + 594331344450528*a(n - 19) - 825476563250976*a(n - 20) + 1043293124084592*a(n - 21) - 1201650502768408*a(n - 22) + 1262594519234968*a(n - 23) - 1210928179506120*a(n - 24) + 1060266691901408*a(n - 25) - 847323181595664*a(n - 26) + 617639581793392*a(n - 27) - 410205458302944*a(n - 28) + 247843724510640*a(n - 29) - 135949707500048*a(n - 30) + 67526545242016*a(n - 31) - 30273460576096*a(n - 32) + 12201462236512*a(n - 33) - 4399521714368*a(n - 34) + 1410734015840*a(n - 35) - 399326676032*a(n - 36) + 98870585152*a(n - 37) - 21165129088*a(n - 38) + 3859085248*a(n - 39) - 587513600*a(n - 40) + 72658304*a(n - 41) - 7011968*a(n - 42) + 495360*a(n - 43) - 22784*a(n - 44) + 512*a(n - 45). - Vaclav Kotesovec, Oct 14 2017

A293587 Number of compositions of n where each part i is marked with a word of length i over a denary alphabet whose letters appear in alphabetical order and all ten letters occur at least once in the composition.

Original entry on oeis.org

102247563, 8624400680, 408962920820, 14395560938040, 419691762832900, 10733397639516016, 249286917950186760, 5378992003398157520, 109550762660946047540, 2130231901794898870880, 39890088439337327537706, 724087830188007677450600, 12806950694169650253597100

Offset: 10

Alois P. Heinz, Oct 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..851

Column k=10 of A261781.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(10):
seq(a(n), n=10..30);

a(n) = 110*a(n-1) - 5610*a(n-2) + 176880*a(n-3) - 3881988*a(n-4) + 63363036*a(n-5) - 803190784*a(n-6) + 8158333238*a(n-7) - 68032529026*a(n-8) + 474993355914*a(n-9) - 2822235496730*a(n - 10) + 14467586756760*a(n - 11) - 64737065451880*a(n - 12) + 255368816478596*a(n - 13) - 895592944790280*a(n - 14) + 2812645592347868*a(n - 15) - 7959012851067608*a(n - 16) + 20400177554223892*a(n - 17) - 47577190249945824*a(n - 18) + 101351234640525316*a(n - 19) - 197858458654518512*a(n - 20) + 354970398396888856*a(n - 21) - 586639546887371480*a(n - 22) + 894863479752319328*a(n - 23) - 1262018115661289704*a(n - 24) + 1647713711756348440*a(n - 25) - 1993736153901444400*a(n - 26) + 2237552288722011272*a(n - 27) - 2330463862262027344*a(n - 28) + 2253297285769248336*a(n - 29) - 2022772844930193632*a(n - 30) + 1685689150486091056*a(n - 31) - 1303653883506384160*a(n - 32) + 935094847660607024*a(n - 33) - 621597594038060528*a(n - 34) + 382531198553819968*a(n - 35) - 217648454420883104*a(n - 36) + 114307777283928640*a(n - 37) - 55307833610580384*a(n - 38) + 24597346495674400*a(n - 39) - 10027630547676256*a(n - 40) + 3735272463460864*a(n - 41) - 1266527133905728*a(n - 42) + 389159192308096*a(n - 43) - 107781232918912*a(n - 44) + 26735152254272*a(n - 45) - 5893548603520*a(n - 46) + 1143628773376*a(n - 47) - 193030560256*a(n - 48) + 27910311552*a(n - 49) - 3387984128*a(n - 50) + 335821568*a(n - 51) - 26104576*a(n - 52) + 1492480*a(n - 53) - 55808*a(n - 54) + 1024*a(n - 55). - Vaclav Kotesovec, Oct 14 2017

cp's OEIS Frontend

A293582 Number of compositions of n where each part i is marked with a word of length i over a quinary alphabet whose letters appear in alphabetical order and all five letters occur at least once in the composition.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A293583 Number of compositions of n where each part i is marked with a word of length i over a senary alphabet whose letters appear in alphabetical order and all six letters occur at least once in the composition.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A293584 Number of compositions of n where each part i is marked with a word of length i over a septenary alphabet whose letters appear in alphabetical order and all seven letters occur at least once in the composition.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A293585 Number of compositions of n where each part i is marked with a word of length i over an octonary alphabet whose letters appear in alphabetical order and all eight letters occur at least once in the composition.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A293586 Number of compositions of n where each part i is marked with a word of length i over a nonary alphabet whose letters appear in alphabetical order and all nine letters occur at least once in the composition.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A293587 Number of compositions of n where each part i is marked with a word of length i over a denary alphabet whose letters appear in alphabetical order and all ten letters occur at least once in the composition.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula