A196930 -id:A196930

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A196931

Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n.

+10
9

0, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,4`
	COMMENTS	`If n >= 1, row n lists the smallest parts of every partition of n in the order produced by the shell model of partitions of A135010, hence row n lists the parts of the last section of the set of partitions of n, except the emergent parts (See A182699).` `Row n has length A000041(n). Row sums give A046746. Right border of triangle gives A001477. Row n starts with A000041(n-1) ones, n >= 1.`
	LINKS	`Table of n, a(n) for n=0..88.`
	EXAMPLE	`Written as a triangle:` `0,` `1,` `1,2,` `1,1,3,` `1,1,1,2,4,` `1,1,1,1,1,2,5,` `1,1,1,1,1,1,1,2,2,3,6` `1,1,1,1,1,1,1,1,1,1,1,2,2,3,7,` `1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,8,` `...`
	CROSSREFS	`Mirror of triangle A182715.` `Cf. A000041, A046746, A135010, A138121, A141285, A182699, A183152, A193827, A196930.`
	KEYWORD	`nonn,tabf`
	AUTHOR	`Omar E. Pol, Oct 21 2011`
	STATUS	`approved`

A196025

Total sum of parts greater than 1 in all the partitions of n except one copy of the smallest part greater than 1 of every partition.

+10
5

0, 0, 0, 2, 5, 16, 30, 63, 108, 189, 298, 483, 720, 1092, 1582, 2297, 3225, 4551, 6244, 8592, 11590, 15622, 20741, 27536, 36066, 47198, 61150, 79077, 101391, 129808, 164934, 209213, 263745, 331807, 415229, 518656, 644719, 799926, 988432, 1218979 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,4`
	COMMENTS	`Also partial sums of A182709. Total sum of emergent parts in all partitions of all numbers <= n.` `Also total sum of parts of all regions of n that do not contain 1 as a part (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012`
	LINKS	`Vaclav Kotesovec, Table of n, a(n) for n = 1..10000`
	FORMULA	`a(n) = A066186(n) - A196039(n).` `a(n) ~ exp(Pisqrt(2n/3)) / (4*sqrt(3)). - Vaclav Kotesovec, Jul 06 2019`
	CROSSREFS	`Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196039, A196930, A196931, A198381.`
	KEYWORD	`nonn`
	AUTHOR	`Omar E. Pol, Oct 27 2011`
	STATUS	`approved`

A198381

Total number of parts greater than 1 in all partitions of n minus the number of partitions of n into parts each less than n.

+10
5

0, 0, 0, 0, 1, 2, 6, 10, 20, 32, 54, 81, 128, 184, 273, 385, 549, 754, 1048, 1412, 1917, 2547, 3392, 4444, 5837, 7556, 9791, 12553, 16086, 20429, 25935, 32665, 41108, 51404, 64190, 79721, 98882, 122043, 150417, 184618, 226239 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,6`
	COMMENTS	`Also partial sums of A182699. Total number of emergent parts in all partitions of the numbers <= n.` `Also total number of parts of all regions of n that do not contain 1 as a part (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012`
	LINKS	`Table of n, a(n) for n=0..40.`
	FORMULA	`a(n) = A096541(n) - A000065(n) = 1 + A096541(n) - A000041(n) = 1 + A006128(n) - A000070(n).` `a(n) = A006128(n) - A026905(n), n >= 1.`
	CROSSREFS	`Cf. A000041, A000065, A000070, A006128, A026905, A093694, A096541, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196930, A196931.`
	KEYWORD	`nonn`
	AUTHOR	`Omar E. Pol, Oct 27 2011`
	STATUS	`approved`

A196039

Total sum of the smallest part of every partition of every shell of n.

+10
2

0, 1, 4, 9, 18, 30, 50, 75, 113, 162, 231, 318, 441, 593, 798, 1058, 1399, 1824, 2379, 3066, 3948, 5042, 6422, 8124, 10264, 12884, 16138, 20120, 25027, 30994, 38312, 47168, 57955, 70974, 86733, 105676, 128516, 155850, 188644, 227783, 274541 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	COMMENTS	`Partial sums of A046746.` `Total sum of parts of all regions of n that contain 1 as a part. - Omar E. Pol, Mar 11 2012`
	LINKS	`Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)` `Omar E. Pol, Illustration of the seven regions of 5`
	FORMULA	`a(n) = A066186(n) - A196025(n).` `a(n) ~ exp(Pisqrt(2n/3)) / (2Pisqrt(2*n)). - Vaclav Kotesovec, Jul 06 2019`
	EXAMPLE	`For n = 5 the seven partitions of 5 are:` `5` `3 + 2` `4 + 1` `2 + 2 + 1` `3 + 1 + 1` `2 + 1 + 1 + 1` `1 + 1 + 1 + 1 + 1` `.` `The five shells of 5 (see A135010 and also A138121), written as a triangle, are:` `1` `2, 1` `3, 1, 1` `4, (2, 2), 1, 1, 1` `5, (3, 2), 1, 1, 1, 1, 1` `.` `The first "2" of row 4 does not count, also the "3" of row 5 does not count, so we have:` `1` `2, 1` `3, 1, 1` `4, 2, 1, 1, 1` `5, 2, 1, 1, 1, 1, 1` `.` `thus a(5) = 1+2+1+3+1+1+4+2+1+1+1+5+2+1+1+1+1+1 = 30.`
	MAPLE	`b:= proc(n, i) option remember;` `if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(n<i, 0, b(n-i, i))) `end:` `a:= proc(n) option remember;` b(n, n) +`if`(n=0, 0, a(n-1)) `end:` `seq(a(n), n=0..50); # Alois P. Heinz, Apr 03 2012`
	MATHEMATICA	`b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i-1] + If[n < i, 0, b[n-i, i]]]; Accumulate[Table[b[n, n], {n, 0, 50}]] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)`
	CROSSREFS	`Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196025, A196930, A196931, A198381, A206437.`
	KEYWORD	`nonn`
	AUTHOR	`Omar E. Pol, Oct 27 2011`
	STATUS	`approved`

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