A210954 -id:A210954

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A210964

Column 10 of square array A195825. Also column 1 of triangle A210954. Also 1 together with the row sums of triangle A210954.

+20
16

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 194, 195 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,12`
	COMMENTS	`Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13, 13], [35, 35, 35, 35, 35], [86, 86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012`
	LINKS	`Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..3000 from Vaclav Kotesovec)` `Vaclav Kotesovec, Graph - The asymptotic ratio` `Michael Somos, Introduction to Ramanujan theta functions` `Eric Weisstein's World of Mathematics, Ramanujan Theta Functions`
	FORMULA	`Expansion of 1 / f(-x, -x^11) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 10 2015` `Partitions of n into parts of the form 12k, 12k+1, 12k+11. - Michael Somos, Jan 10 2015` `Euler transform of period 12 sequence [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, ...]. - Michael Somos, Jan 10 2015` `G.f.: Product_{k>0} 1 / ((1 - x^(12k)) * (1 - x^(12k - 1)) (1 - x^(12k - 11))).` `Convolution inverse of A247133.` `a(n) ~ sqrt(2)(1+sqrt(3)) * exp(Pisqrt(n/6)) / (8n). - Vaclav Kotesovec, Nov 08 2015` `a(n) = (1/n)Sum_{k=1..n} A284372(k)a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017` `a(n) = a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - - (with the convention a(n) = 0 for negative n), where 1, 11, 14, 34, ... is the sequence of generalized 14-gonal numbers A195818. - Peter Bala, Dec 10 2020`
	MATHEMATICA	`nmax = 100; CoefficientList[Series[Product[1 / ((1 - x^(12k)) (1 - x^(12k-1)) (1 - x^(12k-11))), {k, 1, nmax}], {x, 0, nmax}], x] ( Vaclav Kotesovec, Nov 08 2015 *)`
	PROG	`(GW-BASIC)' A program with two A-numbers:` `10 Dim A195818(100), A057077(100), a(100): a(0)=1` `20 For n = 1 to 67: For j = 1 to n` `30 If A195818(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A195818(j))` `40 Next j: Print a(n-1); : Next n` `50 End`
	CROSSREFS	`Cf. A000041, A006950, A036820, A057077, A195818, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A210944, A210954, A211970, A211971.` `Cf. A247133.`
	KEYWORD	`nonn`
	AUTHOR	`Omar E. Pol, Jun 16 2012`
	STATUS	`approved`

A195825

Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem.

+10
38

1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 7, 3, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 15, 5, 3, 1, 1, 1, 1, 1, 22, 7, 4, 2, 1, 1, 1, 1, 1, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1, 56, 16, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 77, 21, 10, 4 (list; table; graph; refs; listen; history; text; internal format)

	OFFSET	`0,4`
	COMMENTS	`In the infinite square array the column k is related to the generalized m-gonal numbers, where m = k+4. For example: the first column is related to the generalized pentagonal numbers A001318. The second column is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on ... (see the program in which A195152 is a table of generalized m-gonal numbers).` `In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041 (see below the first row of the table):` `========================================================` `. Column k of` `. this square` `. Generalized Triangle Triangle array A195825` `k m m-gonal "A" "B" [row sums of` `. numbers triangle "B"` `. with a(0)=1]` `========================================================` `1 5 A001318 A195310 A175003 A000041` `2 6 A000217 A195826 A195836 A006950` `3 7 A085787 A195827 A195837 A036820` `4 8 A001082 A195828 A195838 A195848` `5 9 A118277 A195829 A195839 A195849` `6 10 A074377 A195830 A195840 A195850` `7 11 A195160 A195831 A195841 A195851` `8 12 A195162 A195832 A195842 A195852` `9 13 A195313 A195833 A195843 A196933` `10 14 A195818 A210944 A210954 A210964` `...` `It appears that column 2 of the square array is A006950.` `It appears that column 3 of the square array is A036820.` Conjecture: if k is odd then column k contains (k+1)/2 plateaus whose levels are the first (k+1)/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 2. Otherwise, if k is even then column k contains k/2 plateaus whose levels are the first k/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 3. The sequence A210843 gives the levels of the plateaus of column k, when k -> infinity. For the visualization of the plateaus see the graph of a column, for example see the graph of A210964. - Omar E. Pol, Jun 21 2012
	LINKS	`Table of n, a(n) for n=0..81.` `Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium` `Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.` `Eric Weisstein's World of Mathematics, Pentagonal Number Theorem` `Wikipedia, Pentagonal number theorem`
	FORMULA	`Column k is asymptotic to exp(Pisqrt(2n/(k+2))) / (8sin(Pi/(k+2))n). - Vaclav Kotesovec, Aug 14 2017`
	EXAMPLE	`Array begins:` `1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...` `1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...` `2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...` `3, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...` `5, 3, 2, 1, 1, 1, 1, 1, 1, 1, ...` `7, 4, 3, 2, 1, 1, 1, 1, 1, 1, ...` `11, 5, 4, 3, 2, 1, 1, 1, 1, 1, ...` `15, 7, 4, 4, 3, 2, 1, 1, 1, 1, ...` `22, 10, 5, 4, 4, 3, 2, 1, 1, 1, ...` `30, 13, 7, 4, 4, 4, 3, 2, 1, 1, ...` `42, 16, 10, 5, 4, 4, 4, 3, 2, 1, ...` `56, 21, 12, 7, 4, 4, 4, 4, 3, 2, ...` `77, 28, 14, 10, 5, 4, 4, 4, 4, 3, ...` `101, 35, 16, 12, 7, 4, 4, 4, 4, 4, ...` `135, 43, 21, 13, 10, 5, 4, 4, 4, 4, ...` `176, 55, 27, 14, 12, 7, 4, 4, 4, 4, ...` `...` `Column 1 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11, ... The column contains only one plateau: [1, 1] which has level 1 and length 2.` `Column 3 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10, ... The column contains two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2.` `Column 6 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21, ... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3.`
	PROG	`(GW-BASIC)' A program (with two A-numbers) for the table of example section.` `10 DIM A057077(100), A195152(15, 10), T(15, 10)` `20 FOR k = 1 TO 10 'Column 1-10` `30 T(0, k) = 1 'Row 0` `40 FOR n = 1 TO 15 'Rows 1-15` `50 FOR j = 1 TO n` `60 IF A195152(j, k) <= n THEN T(n, k) = T(n, k) + A057077(j-1) * T(n - A195152(j, k), k)` `70 NEXT j` `80 NEXT n` `90 NEXT k` `100 FOR n = 0 TO 15` `110 FOR j = 1 TO 10` `120 PRINT T(n, k);` `130 NEXT k` `140 PRINT` `150 NEXT n` `160 END` `170 'Omar E. Pol, Jun 18 2012`
	CROSSREFS	`Columns (1-10): A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.` `For another version see A211970.` `Cf. A057077, A195152, A210843.`
	KEYWORD	`nonn,tabl`
	AUTHOR	`Omar E. Pol, Sep 24 2011`
	STATUS	`approved`

A211970

Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

+10
13

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 1, 1, 1, 10, 5, 2, 1, 1, 1, 16, 7, 3, 1, 1, 1, 1, 24, 11, 4, 2, 1, 1, 1, 1, 36, 15, 5, 3, 1, 1, 1, 1, 1, 54, 22, 7, 4, 2, 1, 1, 1, 1, 1, 78, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 112, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)

	OFFSET	`0,4`
	COMMENTS	`In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...` `In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):` `========================================================` `. Column k of` `. this square` `. Generalized Triangle Triangle array A211970` `k m m-gonal "A" "B" [row sums of` `. numbers triangle "B"` `. (if k>=1) with a(0)=1,` `. if k >= 0]` `========================================================` `0 4 A008794 - - A211971` `1 5 A001318 A195310 A175003 A000041` `2 6 A000217 A195826 A195836 A006950` `3 7 A085787 A195827 A195837 A036820` `4 8 A001082 A195828 A195838 A195848` `5 9 A118277 A195829 A195839 A195849` `6 10 A074377 A195830 A195840 A195850` `7 11 A195160 A195831 A195841 A195851` `8 12 A195162 A195832 A195842 A195852` `9 13 A195313 A195833 A195843 A196933` `10 14 A195818 A210944 A210954 A210964` `...` `It appears that column 2 of the square array is A006950.` `It appears that column 3 of the square array is A036820.` `The partial sums of column 0 give A015128. - Omar E. Pol, Feb 09 2014`
	LINKS	`Table of n, a(n) for n=0..77.` `L. Euler, De mirabilibus proprietatibus numerorum pentagonalium` `L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.` `Eric Weisstein's World of Mathematics, Pentagonal Number Theorem`
	FORMULA	`T(n,k) = A211971(n), if k = 0.` `T(n,k) = A195825(n,k), if k >= 1.`
	EXAMPLE	`Array begins:` `1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...` `1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...` `2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...` `4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...` `6, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, ...` `10, 7, 4, 3, 2, 1, 1, 1, 1, 1, 1, ...` `16, 11, 5, 4, 3, 2, 1, 1, 1, 1, 1, ...` `24, 15, 7, 4, 4, 3, 2, 1, 1, 1, 1, ...` `36, 22, 10, 5, 4, 4, 3, 2, 1, 1, 1, ...` `54, 30, 13, 7, 4, 4, 4, 3, 2, 1, 1, ...` `78, 42, 16, 10, 5, 4, 4, 4, 3, 2, 1, ...` `112, 56, 21, 12, 7, 4, 4, 4, 4, 3, 2, ...` `160, 77, 28, 14, 10, 5, 4, 4, 4, 4, 3, ...` `224, 101, 35, 16, 12, 7, 4, 4, 4, 4, 4, ...` `312, 135, 43, 21, 13, 10, 5, 4, 4, 4, 4, ...` `432, 176, 55, 27, 14, 12, 7, 4, 4, 4, 4, ...` `...`
	PROG	`(GW-BASIC)' A program (with two A-numbers) for the square array of the example section.` `10 DIM A057077(100), A195152(15, 10), T(15, 10)` `20 FOR K = 0 TO 10 'Column 0-10` `30 T(0, K) = 1 'Row 0` `40 FOR N = 1 TO 15 'Rows 1-15` `50 FOR J = 1 TO N` `60 IF A195152(J, K) <= N THEN T(N, K) = T(N, K) + A057077(J-1) * T(N - A195152(J, K), K)` `70 NEXT J` `80 NEXT N` `90 NEXT K` `100 FOR N = 0 TO 15: FOR K = 0 TO 10` `110 PRINT T(N, K);` `120 NEXT K: PRINT: NEXT N` `130 END`
	CROSSREFS	`Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.` `For another version see A195825.` `Cf. A057077, A195152.`
	KEYWORD	`nonn,tabl`
	AUTHOR	`Omar E. Pol, Jun 10 2012`
	STATUS	`approved`

A210944

Triangle read by rows with T(n,k) = n - A195818(k), n>=1, k>=1, if (n - A195818(k))>=0.

+10
4

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 11, 1, 12, 2, 13, 3, 0, 14, 4, 1, 15, 5, 2, 16, 6, 3, 17, 7, 4, 18, 8, 5, 19, 9, 6, 20, 10, 7, 21, 11, 8, 22, 12, 9, 23, 13, 10, 24, 14, 11, 25, 15, 12, 26, 16, 13, 27, 17, 14, 28, 18, 15, 29, 19, 16, 30, 20, 17, 31, 21, 18 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,3`
	COMMENTS	`Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195818(k).` `This sequence is related to the generalized 14-gonal numbers A195818, A210954 and A210964 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.`
	LINKS	`Table of n, a(n) for n=1..73.`
	EXAMPLE	`Written as an irregular triangle:` `0;` `1;` `2;` `3;` `4;` `5;` `6;` `7;` `8;` `9;` `10, 0;` `11, 1;` `12, 2;` `13, 3, 0;` `14, 4, 1;` `15, 5, 2;` `16, 6, 3;` `17, 7, 4;` `18, 8, 5;` `19, 9, 6;`
	CROSSREFS	`Cf. A195310, A195825, A195826, A195827, A195828, A195829, A195830, A195831, A195832, A195833, A195818, A210954, A210964.`
	KEYWORD	`nonn,tabf`
	AUTHOR	`Omar E. Pol, Jun 16 2012`
	STATUS	`approved`

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