keyword:new

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Displaying 1-10 of 576 results found.

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A289694

The number of partitions of [n] with exactly 4 blocks without peaks.

+0
0

0, 0, 0, 1, 4, 16, 64, 236, 818, 2736, 8934, 28622, 90324, 281792, 871556, 2677750, 8184383, 24913238, 75593383, 228793147, 691094857, 2084237036, 6277871658, 18890568921, 56798001639, 170665733660, 512554832309, 1538718547049 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,5`
	LINKS	`Table of n, a(n) for n=1..28.` `T. Mansour, M. Shattuck, Counting Peaks and Valleys in a Partition of a Set , J. Int. Seq. 13 (2010), 10.6.8, Table 1.`
	KEYWORD	`nonn,new`
	AUTHOR	`R. J. Mathar, Jul 09 2017`
	STATUS	`approved`

A289693

The number of partitions of [n] with exactly 3 blocks without peaks.

+0
0

0, 0, 1, 3, 9, 27, 75, 197, 503, 1263, 3132, 7695, 18784, 45649, 110585, 267276, 644907, 1554208, 3742321, 9005265, 21659603, 52078400, 125186565, 300870586, 723010749, 1737273406, 4174084259, 10028409724, 24092769583, 57880137331 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,4`
	LINKS	`Table of n, a(n) for n=1..30.` `T. Mansour, M. Shattuck, Counting Peaks and Valleys in a Partition of a Set , J. Int. Seq. 13 (2010), 10.6.8, Table 1.`
	KEYWORD	`nonn,new`
	AUTHOR	`R. J. Mathar, Jul 09 2017`
	STATUS	`approved`

A289692

The number of partitions of [n] with exactly 2 blocks without peaks.

+0
0

0, 1, 2, 4, 8, 15, 27, 48, 85, 150, 264, 464, 815, 1431, 2512, 4409, 7738, 13580, 23832, 41823, 73395, 128800, 226029, 396654, 696080, 1221536, 2143647, 3761839, 6601568, 11584945 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,3`
	LINKS	`Table of n, a(n) for n=1..30.` `T. Mansour, M. Shattuck, Counting Peaks and Valleys in a Partition of a Set , J. Int. Seq. 13 (2010), 10.6.8, Table 1.`
	KEYWORD	`nonn,new`
	AUTHOR	`R. J. Mathar, Jul 09 2017`
	STATUS	`approved`

A289684

Mixing moments for the waiting time in a M/G/1 waiting queue.

+0
0

1, 2, 9, 42, 199, 950, 4554, 21884, 105323, 507398, 2446022, 11796884, 56912838, 274630876, 1325431956, 6397576888, 30882340531, 149084312006, 719736965358, 3474807470756, 16776410481266, 80998307687668, 391074406408716, 1888199373821896, 9116752061308798 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Table of n, a(n) for n=0..24.` `J. Abate, W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, eq. (30) and (32).`
	FORMULA	`G.f. 1/(2-A000108(x)^2), where A000108(x) is the generating function of the Catalan Numbers.` `Conjecture: na(n) +2(-4n+3)a(n-1) +12(n-2)a(n-2) +8(2n-3)*a(n-3)=0.`
	KEYWORD	`nonn,easy,new`
	AUTHOR	`R. J. Mathar, Jul 09 2017`
	STATUS	`approved`

A289683

Mixing moments of the busy period of mean steady-state 1/2 in an M/M/1 waiting process.

+0
0

1, 1, 3, 18, 171, 2250, 37935, 780570, 18967095, 531545490, 16877619675, 598825908450, 23479803807075, 1008211866111450, 47052981361160775, 2371481399995958250, 128370589834339227375, 7427764736129937449250, 457497972176819368669875 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	LINKS	`Table of n, a(n) for n=0..18.` `J. Abate, W. Whitt, Integer Sequences from Queueing Theory , J. Int. Seq. 13 (2010), 10.5.5, top of page 8.`
	FORMULA	`a(n) = n!b(n) where b(0)=1 and b(n) = Sum_{k=0..n-1}binomial(n-1+k, n-1-k) A000108(k) (1/2)^k. [Abate, Eq. 15]` `Conjecture: a(n) +2(-2n+3)a(n-1) +(n-1)(n-3)*a(n-2)=0. - R. J. Mathar, Jul 09 2017`
	KEYWORD	`nonn,easy,new`
	AUTHOR	`R. J. Mathar, Jul 09 2017`
	STATUS	`approved`

A289682

Catalan numbers read modulo 16.

+0
0

1, 1, 2, 5, 14, 10, 4, 13, 6, 14, 12, 2, 12, 4, 8, 13, 6, 6, 12, 6, 4, 12, 8, 2, 12, 12, 8, 4, 8, 8, 0, 13, 6, 6, 12, 14, 4, 12, 8, 6, 4, 4, 8, 12, 8, 8, 0, 2, 12, 12, 8, 12, 8, 8, 0, 4, 8, 8, 0, 8, 0, 0, 0, 13, 6, 6, 12, 14, 4, 12, 8, 14, 4, 4, 8, 12, 8, 8, 0, 6, 4, 4, 8, 4, 8, 8, 0, 12 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	LINKS	`Table of n, a(n) for n=0..87.` `S-C Liu, J. C.-C. Yeh, Catalan numbers modulo 2^k, J. Int. Seq. 13 (2010), 10.5.4, Theorem 5.5.`
	FORMULA	`a(n) = A000108(n) mod 16.`
	MAPLE	`seq ( modp(A000108(n), 16), n=0..120) ;`
	PROG	`(PARI) a(n) = (binomial(2*n, n)/(n+1)) % 16; \\ Michel Marcus, Jul 09 2017`
	CROSSREFS	`Cf. A000108, A036987 (mod 2), A073267 (mod 4), A159987 (mod 8).` `Cf. A048881 (2-adic valuation of A000108).`
	KEYWORD	`nonn,easy,new`
	AUTHOR	`R. J. Mathar, Jul 09 2017`
	STATUS	`approved`

A289654

Related to number of mesh patterns of length 2 that avoid the pattern 321.

+0
0

1, 1, 1, 3, 13, 40, 130, 427, 1428, 4860, 16794 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,4`
	LINKS	`Table of n, a(n) for n=1..11.` `Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.`
	CROSSREFS	`All of A000108, A001453, A246604, A273526, A120304, A289615, A289616, A289652, A289653, A289654 are very similar sequences.`
	KEYWORD	`nonn,more,new`
	AUTHOR	`N. J. A. Sloane, Jul 09 2017`
	STATUS	`approved`

A289653

Related to number of mesh patterns of length 2 that avoid the pattern 321.

+0
0

1, 1, 1, 4, 12, 40, 130, 427, 1428, 4860, 16794 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,4`
	LINKS	`Table of n, a(n) for n=1..11.` `Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.`
	CROSSREFS	`All of A000108, A001453, A246604, A273526, A120304, A289615, A289616, A289652, A289653, A289654 are very similar sequences.`
	KEYWORD	`nonn,more,new`
	AUTHOR	`N. J. A. Sloane, Jul 09 2017`
	STATUS	`approved`

A289652

Related to number of mesh patterns of length 2 that avoid the pattern 321.

+0
0

1, 1, 1, 3, 12, 40, 130, 427, 1428, 4860, 16794 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,4`
	LINKS	`Table of n, a(n) for n=1..11.` `Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.`
	CROSSREFS	`All of A000108, A001453, A246604, A273526, A120304, A289615, A289616, A289652, A289653, A289654 are very similar sequences.`
	KEYWORD	`nonn,more,new`
	AUTHOR	`N. J. A. Sloane, Jul 09 2017`
	STATUS	`approved`

A289616

A246604 (Catalan(n)-n) with initial terms 1,0,0,2,10 changed to 1,1,1,2,11.

+0
0

1, 1, 1, 2, 11, 37, 126, 422, 1422, 4853, 16786, 58775, 208000, 742887, 2674426, 9694830, 35357654, 129644773, 477638682, 1767263171, 6564120400, 24466266999, 91482563618, 343059613627, 1289904147300, 4861946401427, 18367353072126, 69533550915977, 263747951750332, 1002242216651339 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,4`
	COMMENTS	`Related to number of mesh patterns of length 2 that avoid the pattern 321.`
	LINKS	`Table of n, a(n) for n=1..30.` `Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.`
	CROSSREFS	`A variant of A246604.` `All of A000108, A001453, A246604, A273526, A120304, A289615, A289616, A289652, A289653, A289654 are very similar sequences.`
	KEYWORD	`nonn,new`
	AUTHOR	`N. J. A. Sloane, Jul 09 2017`
	STATUS	`approved`

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Last modified July 9 21:22 EDT 2017. Contains 289283 sequences.