A027441 -id:A027441 - cp's OEIS frontend

A068255 1/5 the number of colorings of an n X n square array with 5 colors.

1, 52, 28564, 165770032, 10164078082036, 6584229526795818280, 45062665956031451017237456, 3258395057698765483724093981321824, 2489232886416012985921659124731697904597044, 20091032492258710696689787524926465967570325433558752

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..14

Main diagonal of A222144.

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

A222144 = Cases[Import["https://oeis.org/A222144/b222144.txt", "Table"], {, }][[All, 2]];
a[n_] := A222144[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Sep 23 2019 *)

a(8)-a(10) from Alois P. Heinz, Apr 27 2012

A068256 1/6 the number of colorings of an n X n square array with 6 colors.

Original entry on oeis.org

1, 105, 194485, 6354787485, 3662978221194885, 37246546285522069805565, 6681224184095576349599961437005, 21141920893108925844961568245788270386085, 1180188030501408210062775052100916976604905321333565, 1162187850685436026547128866816039344195930156602955871508107885

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..13

Main diagonal of A222281.

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

A222281 = Cases[Import["https://oeis.org/A222281/b222281.txt", "Table"], {, }][[All, 2]];
a[n_] := A222281[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Sep 23 2019 *)

a(8)-a(10) from Alois P. Heinz, Apr 27 2012

A068257 1/7 the number of colorings of an n X n square array with 7 colors.

Original entry on oeis.org

1, 186, 923526, 122408393436, 433110977725751106, 40908457493732914322944536, 103146129375410533061371714364918916, 6942544711174164051575906086886643368922134556, 12474132532762777585883439690925675118905860580968258566406

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..13

Main diagonal of A222340.

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

A222340 = Cases[Import["https://oeis.org/A222340/b222340.txt", "Table"], {, }][[All, 2]];
a[n_] := A222340[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Sep 23 2019 *)

a(7)-a(9) from Alois P. Heinz, Apr 27 2012

A068258 1/8 the number of colorings of an n X n square array with 8 colors.

Original entry on oeis.org

1, 301, 3418807, 1465295106499, 23698346512668445387, 14462834689097706163375677127, 333066712033498255371201983520013525951, 289435280548175417311368841643540798029239265418611, 9491047284937011500293532002379383630495589849878668222747216079

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..12

Main diagonal of A222462.

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

A222462 = Cases[Import["https://oeis.org/A222462/b222462.txt", "Table"], {, }][[All, 2]];
a[n_] := A222462[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Sep 23 2019 *)

a(6)-a(9) from Alois P. Heinz, Apr 27 2012

A167963 a(n) = n*(n^5 + 1)/2.

Original entry on oeis.org

0, 1, 33, 366, 2050, 7815, 23331, 58828, 131076, 265725, 500005, 885786, 1492998, 2413411, 3764775, 5695320, 8388616, 12068793, 17006121, 23522950, 32000010, 42883071, 56689963, 74017956, 95551500, 122070325, 154457901, 193710258, 240945166, 297411675

Offset: 0

N. J. A. Sloane, Dec 11 2009

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Sequences of the form n*(n^m + 1)/2: A001477 (m=0), A000217 (m=1), A006003 (m=2), A027441 (m=3), A021003 (m=4), this sequence (m=5), A168029 (m=6), A168067 (m=7), A168116 (m=8), A168118 (m=9), A168119 (m=10).

[n*(n^5+1)/2: n in [0..40]]; // Vincenzo Librandi, Dec 10 2014

A167963:=n->n*(n^5+1)/2; seq(A167963(n), n=0..100); # Wesley Ivan Hurt, Nov 23 2013

Table[n(n^5+1)/2, {n,0,100}] (* Wesley Ivan Hurt, Nov 23 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,33,366,2050,7815,23331},30] (* Harvey P. Dale, Dec 09 2014 *)
CoefficientList[Series[x (1 + 26 x + 156 x^2 + 146 x^3 + 31 x^4) / (1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2014 *)

[n*(n^5+1)/2 for n in range(41)] # G. C. Greubel, Jan 17 2023

G.f.: x*(1 + 26*x + 156*x^2 + 146*x^3 + 31*x^4)/(1-x)^7. - Vincenzo Librandi, Dec 10 2014

E.g.f.: (1/2)*x*(2 + 31*x + 90*x^2 + 65*x^3 + 15*x^4 + x^5)*exp(x). - G. C. Greubel, Jan 17 2023

A068240 1/2 the number of colorings of a 4 X 4 square array with n colors.

Original entry on oeis.org

1, 3906, 3000366, 414425080, 19064362455, 428429377026, 5861180425996, 55823546748096, 403783634784285, 2353615149832210, 11531349080992026, 48981767072238936, 184656623163700051, 629125059062885490, 1964980839044519640, 5691311662142685376

Offset: 2

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

a:= n-> n*(n-1)*(17493+(-94782+(251492+(-430857+(529770+(-492434 +(355622+(-202160+(90723+(-31939+(8675+(-1762+(253 +(-23+n)*n)*n) *n)*n)*n)*n) *n)*n) *n)*n) *n)*n)*n) /2:
seq(a(n), n=2..30); # Alois P. Heinz, Apr 27 2012

From Alois P. Heinz, Apr 27 2012 (Start)
G.f.: -(2507986*x^14 +349887529*x^13 +12282125725*x^12 +158263444274*x^11 +896159384816*x^10 +2455337616143*x^9 +3417678462327*x^8 +2453922059100*x^7 +895941969162*x^6 +158666067383*x^5 +12424532171*x^4 +363949394*x^3 +2934100*x^2 +3889*x+1)*x^2 / (x-1)^17.
a(n) = n*(n-1)*(n^14 -23*n^13 +253*n^12 -1762*n^11 +8675*n^10 -31939*n^9 +90723*n^8 -202160*n^7 +355622*n^6 -492434*n^5 +529770*n^4 -430857*n^3 +251492*n^2 -94782*n +17493)/2.
(End)

A068241 1/2 the number of colorings of a 5 X 5 square array with n colors.

Original entry on oeis.org

1, 290493, 10221446382, 25410195205090, 10988934663584655, 1515888422040128871, 94793386050673781548, 3330373652089796835972, 75543449548467802433805, 1216257376373886871239985, 14865437328242111405302266, 144907139188443182894343078

Offset: 2

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

a:= n-> n*(n-1)*(-32126211 +(258309564 +(-1033880579 +(2734090327 +(-5348136944 +(8212654097 +(-10260465459 +(10671809555 +(-9383152441 +(7046057296 +(-4548768800 +(2534102852 +(-1219915634 +(507043269 +(-181400963 +(55554975 +(-14442408 +(3148503 +(-565677 +(81675 +(-9123+(741 +(-39+n)*n) *n)*n)*n)*n) *n)*n)*n) *n)*n)*n) *n)*n)*n) *n)*n)*n) *n)*n)*n) *n)*n)/2:
seq(a(n), n=2..30); # Alois P. Heinz, Apr 27 2012

From Alois P. Heinz, Apr 27 2012 (Start)
G.f.: (64045717133*x^23 +99613598986379*x^22 +30122616672179057*x^21 +2905816841816465011*x^20 +116885162434957285435*x^19 +2301461202426082443493*x^18 +24565180390104215669199*x^17 +152051416127748639010437*x^16 +570955972331169762888066*x^15 +1339611184016759341097870*x^14 +1999028208566595454861898*x^13 +1912423825883782158177854*x^12 +1171838449935804166262422*x^11 +455354414964383806296586*x^10 +109981844564513940260830*x^9 +15982890606970244203818*x^8 +1330217331928452928929*x^7 +58885777127277221367*x^6 +1238407862810793461*x^5 +10331590803059615*x^4 +25144532006783*x^3 +10213893889*x^2 +290467*x+1)*x^2 / (x-1)^26.
a(n) = n*(n-1)*(n^23 -39*n^22 +741*n^21 -9123*n^20 +81675*n^19 -565677*n^18 +3148503*n^17 -14442408*n^16 +55554975*n^15 -181400963*n^14 +507043269*n^13 -1219915634*n^12 +2534102852*n^11 -4548768800*n^10 +7046057296*n^9 -9383152441*n^8 +10671809555*n^7 -10260465459*n^6 +8212654097*n^5 -5348136944*n^4 +2734090327*n^3 -1033880579*n^2 +258309564*n -32126211)/2.
(End)

A068242 1/2 the number of colorings of a 6 X 6 square array with n colors.

Original entry on oeis.org

1, 50798448, 190026633752982, 16460573816989545700, 111739638856566209416695, 143179601228065200130305876, 57851338756390824653502708508, 10086461440383360741777407234232, 929834945885124428238498952273725, 52047326332129638504907000521132040

Offset: 2

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Alois P. Heinz, G.f. for A068242

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

More terms from Alois P. Heinz, Apr 27 2012

A068243 1/2 the number of colorings of a 7 X 7 square array with n colors.

Original entry on oeis.org

1, 20934997854, 19278946338342653286, 112656664890078627543093640, 20043672552286729048799884311015, 361011452813936865714801000277216206, 1332266848133993021484807934080054103804, 1550797305138088176220498521209275420942656

Offset: 2

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Alois P. Heinz, Formula for a(n)

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

More terms from Alois P. Heinz, Apr 27 2012

A068246 1/6 the number of colorings of a 5 X 5 rhombic hexagonal array with n colors.

Original entry on oeis.org

1, 672384, 24673292910, 47694893373440, 16222878355401375, 1842996126472816896, 98798500424990038764, 3068393771393664491520, 62960689342002146953005, 933100311834971308336000, 10639781338324232990590266, 97779035968707368095801344, 750090455889142956720814955

Offset: 3

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

a:= n-> (3008737472+ (-26856982336+ (115567646848+ (-319382723824+ (636837385892+ (-975405045160+ (1192546680096+ (-1193738274422+ (995467197535+ (-699933854941+ (418375982241+ (-213720456031+ (93568827565+ (-35133626327+ (11298632622+
(-3101089711+ (722137763+ (-141421592+ (23000726+ (-3051871+ (321994+ (-25992+ (1508+(-56+n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n/6:
seq (a(n), n=3..40); #  Alois P. Heinz, May 02 2012

G.f.: (1155805517421*x^22 +898154715023598*x^21 +153334491715682431*x^20 +9260621966248364140*x^19 +250086793798293779695*x^18 +3463005755473293705486*x^17 +26809839147864527991573*x^16 +122805799859998392511056*x^15 +345417237429621912129330*x^14 +610511151468783633149340*x^13 +686259871966584143669766*x^12 +491767778082675626596168*x^11 +223082415423639038320846*x^10 +62970879259692393145420*x^9 +10739574336476388551610*x^8 +1057138433525073018576*x^7 +56029398700931117553*x^6 +1436637989069258166*x^5 +14990828199704235*x^4 +47053606279980*x^3 +24655811251*x^2+672358*x+1)*x^3 / (x-1)^26. - Alois P. Heinz, May 02 2012

Extended beyond a(10) by Alois P. Heinz, May 02 2012

cp's OEIS Frontend

A068255 1/5 the number of colorings of an n X n square array with 5 colors.

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A068256 1/6 the number of colorings of an n X n square array with 6 colors.

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A068257 1/7 the number of colorings of an n X n square array with 7 colors.

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A068258 1/8 the number of colorings of an n X n square array with 8 colors.

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A167963 a(n) = n*(n^5 + 1)/2.

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A068240 1/2 the number of colorings of a 4 X 4 square array with n colors.

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Maple

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A068241 1/2 the number of colorings of a 5 X 5 square array with n colors.

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Maple

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A068242 1/2 the number of colorings of a 6 X 6 square array with n colors.

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A068243 1/2 the number of colorings of a 7 X 7 square array with n colors.

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