A034261 -id:A034261 - cp's OEIS frontend

A107600 Column 5 of array illustrated in A089574 and related to A034261.

1, 18, 101, 357, 978, 2274, 4711, 8954, 15915, 26806, 43197, 67079, 100932, 147798, 211359, 296020, 406997, 550410, 733381, 964137, 1252118, 1608090, 2044263, 2574414, 3214015, 3980366, 4892733, 5972491, 7243272, 8731118, 10464639

Offset: 9

Alford Arnold, May 17 2005

The sequences in A089574 count ordered partitions. Sequence A001296 can be associated with 9 = 3+3+3. Six times sequence A005585, associated with 10 = 3+3+2+2. The other three sequences comprising A107600 are generated in A034261 and can be associated with 10 = 5 + 5 = 4 + 4 + 2 = 2 + 2 + 2 + 2 + 2.

			A107600(n) can be constructed from five other sequences as follows:
1...7...25...65...140.......A001296
....1...11...56...196.......A034264
....6...42..162...462.......6.*.A005585.
....3...18...60...150.......A006011
....1....5...14....30.......A000330
therefore
1..18..101..357...978.......A107600

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A034261, A089574, A000330, A001296, A005585, A006011, A034264.

a:= n-> `if` (n<9, 0, (92292 +(-6580 +(-5745 +(1535 +(-147+5*n) *n) *n) *n) *n) *n /720 -218): seq(a(n), n=9..45); # Alois P. Heinz, Nov 06 2009

Select[CoefficientList[Series[(x^5-5x^4+7x^3+4x^2-11x-1)x^9/(x-1)^7, {x,0,50}],x],#>0&] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1,18,101,357,978,2274,4711},42] (* Harvey P. Dale, May 01 2011 *)

G.f.: (x^5 -5*x^4 +7*x^3 +4*x^2 -11*x -1) *x^9 /(x-1)^7. - Alois P. Heinz, Nov 06 2009

More terms from Alois P. Heinz, Nov 06 2009

A107601 Column 6 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

7, 71, 350, 1204, 3329, 7936, 16981, 33452, 61719, 107953, 180620, 291056, 454129, 688994, 1019947, 1477384, 2098871, 2930331, 4027354, 5456636, 7297553, 9643876, 12605633, 16311124, 20909095, 26571077, 33493896, 41902360, 52052129

Offset: 0

Alford Arnold, May 27 2005

nonn

Column six is associated with the following partitions: 66, 552, 443, 4422, 3332, 33222 and 222222.

			a(1) = 7 because there are 3 permutations of 443 and 4 permutations of 3332.
a(2) = 71 because there are 1 + 3 + 18 + 6 + 32 + 10 + 1 permutations of the associated partitions.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1). [From _R. J. Mathar_, Jun 26 2010]

Cf. A107600.

a(n) = A107601(10+n,5+n) = n*(1384+4683*n^2+2142*n+2*n^6+651*n^4+63*n^5+2835*n^3)/1680. a(n)= +8*a(n-1) -28*a(n-2) +56*a(n-3) -70*a(n-4) +56*a(n-5) -28*a(n-6) +8*a(n-7) -a(n-8). G.f.: x*(7+15*x-22*x^2+11*x^4-6*x^5+x^6)/(x-1)^8. [From R. J. Mathar, Jun 26 2010]

Extended beyond a(5) by R. J. Mathar, Jun 26 2010

A109125 Column 7 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

2, 41, 292, 1283, 4253, 11712, 28261, 61738, 124763, 236762, 426557, 735616, 1222064, 1965563, 3073176, 4686337, 6989056, 10217495, 14671058, 20725145, 28845727, 39605906, 53704631, 71987748, 95471569, 125369152, 163119491

Offset: 0

Alford Arnold, Jun 19 2005

			a(1) = 2 because 4+4+4 and 3+3+3+3 cannot be permuted. a(2) = 41 because there are 3 + 7 + 12 + 9 + 10 ways of permuting the associated partitions.
5553 (3 ways), 4441 & 544 (4+3 ways), 4432 (12 ways), 33331 & 4333 (5 + 4 ways) and 33322 (in 10 ways).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1). [From _R. J. Mathar_, Jun 26 2010]

Cf. A000124, A000330, A034261, A086602, A089574, A107601.

LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{2,41,292,1283,4253,11712,28261,61738,124763},30] (* Harvey P. Dale, Aug 21 2024 *)

From R. J. Mathar, Jun 26 2010: (Start)
a(n) = A105552(11+n,5+n).
G.f.: x*(-2-23*x+5*x^2+37*x^3-26*x^4-9*x^5+17*x^6-7*x^7+x^8)/(x-1)^9. a(n) = -1+1277*n/840 -19*n^3/480 -67*n^2/480 +41*n^5/80 +257*n^6/2800 +23*n^7/3360 +6049*n^4/5760 +n^8/5760. (End)

Extended beyond a(6) by R. J. Mathar, Jun 26 2010

A108538 Column 10 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

1, 64, 731, 4553, 20155, 71272, 214653, 572743, 1389702, 3122752, 6585183, 13162741, 25131718, 46115029, 81722067, 140429357, 234772177, 382932581, 610826859, 954815625, 1465182669, 2210554686, 3283463257, 4807283267, 6944818576, 9908846494, 13974977743, 19497238421, 26926835328

Offset: 0

Alford Arnold, Jul 05 2005

A109820 can be decomposed into 30 sequences. These 30 associated sequences can be inferred from the 30 ways of partitioning the number nine: 9 81 72 63 54 ... the complete listing is available in the Handbook of Mathematical Functions (1964) p. 831. Consider, for example, the three ways of partitioning the number three: 3, 21 and 111; prepend each partition then add one to each value - yielding 44, 332 and 2222. These "associated" partitions are then used to derive the associated sequences. 44 => A000330, 332 => A006011 and 2222 => A034263. Summing these three sequences yields A089574.

			a(1) = 1 because the only associated partition 4444 for n = 16 cannot be permuted.
a(2) = 64 because the associated partitions can be permuted in 3 + 4 + 12 + 9 + 20 + 10 + 6 ways when n = 17.

Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000330 (column 2), A086602 (column 3), A089574 (column 4), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13)

G.f. 1+64*x -x^2*(-731 +4219*x -13765*x^2 +30910*x^3 -49804*x^4 +58458*x^5 -50237*x^6 +31394*x^7 -13931*x^8 +4171*x^9 -757*x^10 +63*x^11)/(x-1)^12 . - R. J. Mathar, Aug 28 2018

Extended by R. J. Mathar, Aug 28 2018

A109820 Column 9 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

6, 126, 992, 4921, 18450, 57198, 154420, 375106, 838075, 1749221, 3449895, 6485363, 11699374, 20362113, 34340211, 56319046, 90089305, 140911696, 215975810, 324971445, 480793226, 700402096, 1005870222, 1425639066, 1996023823, 2763001135, 3784320961, 5131987727, 6895160406, 9183525995, 12131205973

Offset: 0

Alford Arnold, Jul 03 2005

			The associated sequences begin for n = 15 through 19:
........................1.......5
........................3.......18
................3.......18......60
........3.......18......60......150
1.......7.......25......65......140
........................6.......42
................12......84......324
........12......84......324.....924
........6.......42......162.....462
4.......32......132.....392.....952
........................10......80
................30......240.....1050
........10......90......420.....1400
........30......240.....1050....3360
1.......11......56......196.....546
........................15......135
................60......540.....2640
........15......165.....900.....3420
........................21......210
................35......385.....2205
........................28......308
........................1.......19
therefore this sequence begins
6 126 992 4921 18450

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A109126.

Cf. A000330 (column 2), A086602 (column 3), A089574 (column 4), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13).

G.f. 6 -x*( 126 -394*x +939*x^2 -1911*x^3 +2803*x^4 -2825*x^5 +1964*x^6 -939*x^7 +298*x^8 -57*x^9 +5*x^10)/(x-1)^11 . - R. J. Mathar, Aug 28 2018

More terms with the program of A105552 from R. J. Mathar, Aug 28 2018

A109821 Column 11 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

27, 482, 3855, 20329, 82346, 277295, 813738, 2145712, 5192450, 11708366, 24881487, 50269005, 97217758, 180966915, 325691821, 568823951, 967074547, 1604701323, 2604691419, 4143692621, 6471712062, 9937820779, 15023357512, 22384420182, 32905773076, 47768686720

Offset: 0

Alford Arnold, Jul 18 2005

nonn

			An examination of the relevant ordered Gaussian polynomials reveals the following distribution (beginning with partitions of length three):
1 10 15 1
6 52 180 216 28
12 114 530 1386 1547 266
18 168 880 3086 7007 7616 1554
therefore (by summing each row) this sequence begins
27
482
3855
20329

Georg Fischer, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A034261, A109820.

Cf. A000330 (column 2), A086602 (column 3), A089574 (column 4), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13).

LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1}, {27,482, 3855,20329,82346,277295,813738,2145712,5192450,11708366,24881487, 50269005, 97217758,180966915,325691821}, 1001] (* Georg Fischer, Feb 28 2019 *)

G.f.: 27 + 482*x -x^2*(3855 -29786*x +118759*x^2 -310071*x^3 +574122*x^4 -780978*x^5 +792535*x^6 -601009*x^7 +336759*x^8 -135622*x^9 +37194*x^10 -6228*x^11 +481*x^12) /(x-1)^13. - R. J. Mathar, Aug 28 2018

More terms from R. J. Mathar, Aug 28 2018

A110553 Column 12 of an array illustrated in A089584 and related to A034261.

Original entry on oeis.org

9, 284, 3004, 19078, 88938, 335612, 1084387, 3109060, 8104089, 19539904, 44141520, 94346102, 192252586, 375787005, 708083995, 1291443529, 2287680232, 3947261426, 6650353141, 10963787826, 17719064134, 28117822582, 43872849975, 67394593662, 102035462287, 152406906280

Offset: 0

Alford Arnold, Jul 29 2005

The column sequences can also be calculated using sequences which map to associated partitions. For example, 4 32 132 392 ... maps to 5+5+5+4 (n=19) and sequence 5 50 245 840 ... maps to 4+4+4+4+3. Many partitions map to the same sequences since the mapping depends only on the "degree" of the partition. In the above two cases, the degrees are 31 and 41 respectively. At n = 20 the relevant degrees are: 21,31,211,311,22,221,42,212,321,24 and 61. The associated partitions can be permuted with the number of ways as indicated: 3 4 12 20 6 30 15 30 60 15 and 7 ways. Adding these values with the 32 and 50 ways from our first two sequences confirms that A110553(2) = 284.

			An examination of the relevant ordered Gaussian polynomials reveals the following distributions:
5 4
7 120 120 34 3
112 1127 1190 470 96 9
882 6692 7147 3270 910 162 15
therefore the sequence begins
9
284
3004
19078
...

Georg Fischer, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Cf. A000330 (column 2), A086602 (column 3), A089574 (column 4), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13).

LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1}, {9, 284, 3004,19078, 88938, 335612, 1084387, 3109060, 8104089,19539904, 44141520, 94346102, 192252586, 375787005, 708083995, 1291443529, 2287680232}, 1001] (* Georg Fischer, Feb 28 2019 *)

G.f.: 9+284*x+3004*x^2 -x^3*(-19078 +178154*x -826578*x^2 +2465215*x^3 -5191980*x^4 +8073520*x^5 -9475220*x^6 +8461596*x^7 -5732830*x^8 +2904174*x^9 -1067563*x^10 +269335*x^11 -41760*x^12 +3003*x^13) /(x-1)^14. - R. J. Mathar, Aug 28 2018

More terms from R. J. Mathar, Aug 28 2018

A110624 Column 13 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

2, 148, 2159, 16746, 90371, 383147, 1364679, 4256436, 11952264, 30812675, 73980045, 167235586, 358959637, 736521122, 1452480779, 2765433413, 5102243735, 9150977405, 15997148389, 27320179543, 45672422411, 74869958841, 120533007335, 190824424010, 297447916702, 456983633899, 692658264469, 1036670276472, 1533219097805

Offset: 1

Alford Arnold, Aug 26 2005

nonn

The 77 applicable partitions range from n = 20 through n=26 with the following frequency distribution: 2, 9, 16, 15, 14, 9 and 12.

			A110624(1) = 2 because the two relevant partitions of 20 are 4+4+4+4+4 and 5+5+5+5 and cannot be permuted.
A110624(2) = 148 because at n = 21 the relevant partitions can be permuted in 20 plus 128 = 148 ways.
The 20 ways are apparent from A034261 by observing that 9 + 11 = 20 in the subsequences 1,9,39,... and 1,11,56,...
and the 128 ways result from examining the nine partitions of n = 21 having 3,31,32,33,7,211,221,311 and 411 degrees.
A110624(4)= A110554(13) = 16746.

Cf. A034261, A110554.

Cf. A000330 (column 2), A086602 (column 3), A089574 (column 4), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13)

More terms from R. J. Mathar, Aug 28 2018

A109126 Column 8 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

18, 207, 1198, 4825, 15448, 42168, 102297, 226530, 466357, 904352, 1668083, 2948502, 5023797, 8289819, 13298336, 20804513, 31825172, 47709549, 70224436, 101655775, 144928958, 203750282, 282772211, 387785308, 525939919, 706000918, 938639057, 1236762708, 1615894035, 2094593893

Offset: 0

Alford Arnold, Jun 22 2005

			The associated partitions begin (for n = 14, 15, 16, ...
................1.......5.......14
................3.......18......60
........3.......18......60......150
3.......18......60......150.....315
................6.......42......162
........12......84......324.....924
4.......32......132.....392.....952
6.......42......162.....462.....1092
................10......80......350
........30......240.....1050....3360
5.......50......245.....840.....2310
................15......135.....660
........20......200.....1040....3840
................21......210.....1134
................1.......17......125
therefore this sequence begins
18 207 1198 4825 15448 ...

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A105552, A000330 (column 2), A086602 (column 3), A089574 (column 4), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13).

G.f. ( 18+27*x -62*x^2 +48*x^4 -13*x^5 -27*x^6 +24*x^7 -8*x^8 +x^9 ) / (x-1)^10 . - R. J. Mathar, Aug 28 2018

More terms with the program of A105552 from R. J. Mathar, Aug 28 2018

A034264 a(n) = f(n,4) where f is given in A034261.

Original entry on oeis.org

0, 1, 11, 56, 196, 546, 1302, 2772, 5412, 9867, 17017, 28028, 44408, 68068, 101388, 147288, 209304, 291669, 399399, 538384, 715484, 938630, 1216930, 1560780, 1981980, 2493855, 3111381, 3851316, 4732336, 5775176, 7002776, 8440432

Offset: 0

Clark Kimberling

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A034261.

A034261 := proc(n, k) binomial(n+k, k+1)*(n*k+n+1)/(k+2); end;
seq( A034261(n,4),n=0..40) ;

LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,11,56,196,546,1302},40] (* Harvey P. Dale, Jul 11 2025 *)

G.f.: -x*(1+4*x)/(x-1)^7. - R. J. Mathar, Feb 10 2025

a(n) = n*(5*n+1)*(n+4)*(n+3)*(n+2)*(n+1)/720. - R. J. Mathar, Feb 10 2025

Corrected and extended by N. J. A. Sloane, Apr 21 2000

cp's OEIS Frontend

A107600 Column 5 of array illustrated in A089574 and related to A034261.

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A107601 Column 6 of array illustrated in A089574 and related to A034261.

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A109125 Column 7 of array illustrated in A089574 and related to A034261.

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A108538 Column 10 of array illustrated in A089574 and related to A034261.

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A109820 Column 9 of array illustrated in A089574 and related to A034261.

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A109821 Column 11 of array illustrated in A089574 and related to A034261.

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A110553 Column 12 of an array illustrated in A089584 and related to A034261.

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A110624 Column 13 of array illustrated in A089574 and related to A034261.

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