A002417 -id:A002417 - cp's OEIS frontend

A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 9, 10, 5, 1, 1, 6, 10, 16, 15, 6, 1, 1, 5, 18, 20, 25, 21, 7, 1, 1, 8, 15, 40, 35, 36, 28, 8, 1, 1, 9, 27, 35, 75, 56, 49, 36, 9, 1

Offset: 1

Alford Arnold, Mar 29 2007

The array can be constructed by beginning with A007318 (Pascal's triangle) placing each diagonal on a prime row. The other rows are filled in by mapping the prime factorization of the row number to the known sequences on the prime rows and multiplying term by term.

			Row six begins 1 6 18 40 75 126 ... because rows two and three are
1 2 3 4 5 6 ...
1 3 6 10 15 21 ...
The array begins
1 1 1 1 1 1 1 1 1 A000012
1 2 3 4 5 6 7 8 9 A000027
1 3 6 10 15 21 28 36 45 A000217
1 4 9 16 25 36 49 64 81 A000290
1 4 10 20 35 56 84 120 165 A000292
1 6 18 40 75 126 196 288 405 A002411
1 5 15 35 70 126 210 330 495 A000332
1 8 27 64 125 216 343 512 729 A000578
1 9 36 100 225 441 784 1296 2025 A000537
1 8 30 80 175 336 588 960 1485 A002417
1 6 21 56 126 252 462 792 1287 A000389
1 12 54 160 375 756 1372 2304 3645 A019582
1 7 28 84 210 462 924 1716 3003 A000579
1 10 45 140 350 756 1470 2640 4455 A027800
1 12 60 200 525 1176 2352 4320 7425 A004302
1 16 81 256 625 1296 2401 4096 6561 A000583
1 8 36 120 330 792 1716 3432 6435 A000580
1 18 108 400 1125 2646 5488 10368 18225 A019584
1 9 45 165 495 1287 3003 6435 12870 A000581
1 16 90 320 875 2016 4116 7680 13365 A119771
1 15 90 350 1050 2646 5880 11880 22275 A001297
1 12 63 224 630 1512 3234 6336 11583 A027810
1 10 55 220 715 2002 5005 11440 24310 A000582
1 24 162 640 1875 4536 9604 18432 32805 A019583
1 16 100 400 1225 3136 7056 14400 27225 A001249
1 14 84 336 1050 2772 6468 13728 27027 A027818
1 27 216 1000 3375 9261 21952 46656 91125 A059827
1 20 135 560 1750 4536 10290 21120 40095 A085284

Cf. A000040 A007318.

Cf. A064553 (second diagonal), A080688 (second diagonal resorted).

A128629 := proc(n,m) if n = 1 then 1; elif isprime(n) then p := numtheory[pi](n) ; binomial(p+m-1,p) ; else a := 1 ; for p in ifactors(n)[2] do a := a* procname(op(1,p),m)^ op(2,p) ; od: fi; end: # R. J. Mathar, Sep 09 2009

A-number added to each row of the examples by R. J. Mathar, Sep 09 2009

A151974 a(n) = n(n+1)(n+2)(n+3)(n+4)/8.

Original entry on oeis.org

0, 15, 90, 315, 840, 1890, 3780, 6930, 11880, 19305, 30030, 45045, 65520, 92820, 128520, 174420, 232560, 305235, 395010, 504735, 637560, 796950, 986700, 1210950, 1474200, 1781325, 2137590, 2548665, 3020640, 3560040, 4173840, 4869480, 5654880, 6538455, 7529130

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 21 2009

Also the number of 4-cycles in the (n+3)-triangular graph. - Eric W. Weisstein, Aug 14 2017

Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Johnson Graph.
Eric Weisstein's World of Mathematics, Triangular Graph.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A054559.

Cf. A002417 (number of 3-cycles in the triangular graph), A290939 (5-cycles), A290940 (6-cycles).

A151974:=n->n*(n+1)*(n+2)*(n+3)*(n+4)/8: seq(A151974(n), n=0..60); # Wesley Ivan Hurt, Feb 11 2017

Table[Pochhammer[n, 5]/8, {n, 0, 31}] (* or *)
Rest @ CoefficientList[Series[15 x^2/(1 - x)^6, {x, 0, 32}], x] (* Michael De Vlieger, Feb 12 2017 *)
Pochhammer[Range[0, 20], 5]/8 (* Eric W. Weisstein, Aug 14 2017 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 15, 90, 315, 840, 1890}, 20] (* Eric W. Weisstein, Aug 14 2017 *)
Table[15 Binomial[n + 4, 5], {n, 0, 20}] (* Eric W. Weisstein, Aug 14 2017 *)
15 Binomial[Range[4, 24], 5] (* Eric W. Weisstein, Aug 14 2017 *)
Table[(24 n+50 n^2+35 n^3+10 n^4+n^5)/8,{n,0,40}] (* or *) Table[Times@@Range[n,n+4]/8,{n,0,40}] (* Harvey P. Dale, Mar 06 2024 *)

a(n)=n*(n+1)*(n+2)*(n+3)*(n+4)/8 \\ Charles R Greathouse IV, Aug 14 2017

a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)/8.
G.f.: 15*x/(1-x)^6. - Colin Barker, Jun 25 2012
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Eric W. Weisstein, Aug 14 2017
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = 16*log(2)/3 - 131/36. (End)

Offset corrected by Eric W. Weisstein, Aug 14 2017

A060103 Fourth column (m=3) of triangle A060102.

Original entry on oeis.org

1, 13, 71, 259, 742, 1806, 3906, 7722, 14223, 24739, 41041, 65429, 100828, 150892, 220116, 313956, 438957, 602889, 814891, 1085623, 1427426, 1854490, 2383030, 3031470, 3820635, 4773951, 5917653

Offset: 0

Wolfdieter Lang, Apr 06 2001

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A000012, A000290, A002417, A060102.

Table[((4n^2+20n+15)Binomial[n+4,4])/15,{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{1,13,71,259,742,1806,3906},30] (* Harvey P. Dale, Dec 16 2012 *)

a(n) = (4*n^2+20*n+15)*binomial(n+4, 4)/15.
G.f.: (1+6*x+x^2)/(1-x)^7.
a(0)=1, a(1)=13, a(2)=71, a(3)=259, a(4)=742, a(5)=1806, a(6)=3906, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Dec 16 2012

A068244 1/6 the number of colorings of a 3 X 3 rhombic- or staggered- hexagonal array with n colors.

Original entry on oeis.org

1, 176, 5490, 65600, 455875, 2239776, 8647716, 27962880, 78920325, 200002000, 464447126, 1003294656, 2039332295, 3935444800, 7261533000, 12884914176, 22089914121, 36733221360, 59442494650, 93866696000, 144987663051, 219503536736, 326295822700, 476993088000

Offset: 3

R. H. Hardin, Feb 24 2002

nonn

Numbers for rhombic- and staggered- hexagonal arrays differ above 4 X 4.

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

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

a:= n-> (248 +(-1012 +(1786 +(-1791 +(1120 +(-448 +(112 +(-16+n)*n) *n) *n) *n) *n) *n) *n) *n/6:
seq(a(n), n=3..40);  #  Alois P. Heinz, May 02 2012

From Alois P. Heinz, May 02 2012: (Start)
G.f.: (1089*x^6+10934*x^5+26015*x^4+18500*x^3+3775*x^2+166*x+1) / (x-1)^10*x^3.
a(n) = (n^9 -16*n^8 +112*n^7 -448*n^6 +1120*n^5 -1791*n^4 +1786*n^3 -1012*n^2 +248*n)/6.  (End)

A068245 1/6 the number of colorings of a 4 X 4 rhombic- or staggered- hexagonal array with n colors.

Original entry on oeis.org

1, 7616, 5141250, 552093440, 20631905875, 395001645696, 4771909547076, 41190314035200, 275192443300005, 1502690499112000, 6971521964029766, 28275884687022336, 102456840191225975, 337289521082456320, 1022310183284613000, 2883605488481550336, 7636012822945480521

Offset: 3

R. H. Hardin, Feb 24 2002

Numbers for rhombic- and staggered- hexagonal arrays differ above 4 X 4.

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

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

[(n^11 -26*n^10 +310*n^9 -2240*n^8 +10915*n^7 -37726*n^6 +94576*n^5 -172395*n^4 +224588*n^3 -199854*n^2 +109788*n -28340)*n *(n-1)*(n-2)^3/6: n in [3..19]]; // Bruno Berselli, May 03 2012

a:= n-> (-226720+ (1445104+ (-4304712+ (7968348+ (-10265148+ (9755858+ (-7068408+ (3975561+ (-1749715+ (602408+ (-160859+ (32703+ (-4898+ (510+ (-33+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

From Alois P. Heinz, May 02 2012: (Start)
G.f.: -(7926831*x^13 +710120929*x^12 +16477733814*x^11 +144915014346*x^10 +569769493505*x^9 +1086745824783*x^8 +1040642122932*x^7 +499586289612*x^6 +115866023553*x^5 +11940350895*x^4 +465727286*x^3 +5011914*x^2 +7599*x+1) *x^3 / (x-1)^17.
a(n) = (n^16 -33*n^15 +510*n^14 -4898*n^13 +32703*n^12 -160859*n^11 +602408*n^10 -1749715*n^9  +3975561*n^8 -7068408*n^7 +9755858*n^6 -10265148*n^5 +7968348*n^4 -4304712*n^3 +1445104*n^2 -226720*n)/6. (End)

Extended beyond a(15) by Alois P. Heinz, May 02 2012

A068254 1/4 the number of colorings of an n X n square array with 4 colors.

Original entry on oeis.org

1, 21, 2403, 1500183, 5110723191, 95013316876491, 9639473169171326643, 5336900216006709884938623, 16124704040675904181778734982451, 265865038636937159336134567410478299051

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..16

Main diagonal of A222444.

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

A222444 = Cases[Import["https://oeis.org/A222444/b222444.txt", "Table"], {, }][[All, 2]];
a[n_] := A222444[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Sep 23 2019 *)

a(9)-a(10) from Alois P. Heinz, Apr 27 2012

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

Original entry on oeis.org

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

cp's OEIS Frontend

A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

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A151974 a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)/8.

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A060103 Fourth column (m=3) of triangle A060102.

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A068244 1/6 the number of colorings of a 3 X 3 rhombic- or staggered- hexagonal array with n colors.

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A068245 1/6 the number of colorings of a 4 X 4 rhombic- or staggered- hexagonal array with n colors.

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A068254 1/4 the number of colorings of an n X n square array with 4 colors.

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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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A151974 a(n) = n(n+1)(n+2)(n+3)(n+4)/8.