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A026820
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Euler's table: triangular array T read by rows, where T(n,k) = number of partitions in which every part is <=k for 1<=k<=n. Also number of partitions of n into at most k parts.
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27
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1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 11, 13, 14, 15, 1, 5, 10, 15, 18, 20, 21, 22, 1, 5, 12, 18, 23, 26, 28, 29, 30, 1, 6, 14, 23, 30, 35, 38, 40, 41, 42, 1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56, 1, 7, 19, 34, 47, 58
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| T(T(n,n),n) = A134737(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 07 2007
Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 21 2010: (Start)
Row sums give A058397; central terms give A171985;
T(n,1) = A000012(n);
T(n,2) = A008619(n) for n>1;
T(n,3) = A001399(n) for n>2;
T(n,4) = A001400(n) for n>3;
T(n,5) = A001401(n) for n>4;
T(n,6) = A001402(n) for n>5;
T(n,7) = A008636(n) for n>6;
T(n,8) = A008637(n) for n>7;
T(n,9) = A008638(n) for n>8;
T(n,10) = A008639(n) for n>9;
T(n,11) = A008640(n) for n>10;
T(n,12) = A008641(n) for n>11;
T(n,n-2) = A007042(n-1) for n>2;
T(n,n-1) = A000065(n) for n>1;
T(n,n) = A000041(n). (End)
T(A000217(n),n) = A173519(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 20 2010]
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.
G. Chrystal, Algebra, Vol. II, p. 558.
L. Euler, Introductio in Analysin Infinitorum, Book I, chapter XVI.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.
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LINKS
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences related to partitions [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 21 2010]
Robert G. Wilson v, Table of n, a(n) for n = 1..3240 .
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FORMULA
| T(n,k) = T(n,k-1) + T(n-k,k). [From Thomas Ahle (thomas.ahle(AT)magd.ox.ac.uk), Jun 13 2011]
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EXAMPLE
| 1;
1,2;
1,2,3;
1,3,4,5;
1,3,5,6,7; ...
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MATHEMATICA
| t[n_, k_] := Length@ IntegerPartitions[n, k]; Table[ t[n, k], {n, 12}, {k, n}] // Flatten
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CROSSREFS
| Partial sums of rows of A008284.
Cf. A026840. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 21 2010]
Sequence in context: A036838 A066010 A109974 * A091438 A011794 A073300
Adjacent sequences: A026817 A026818 A026819 * A026821 A026822 A026823
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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