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A274342
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Irregular triangle read by rows, giving the numerators of the coefficients of the Eisenstein series G_{2*n} multiplied by 2*n-1, for n >= 2. Also Laurent coefficients of Weierstrass's P function.
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2
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1, 1, 1, 3, 1, 2, 2, 60, 5, 1, 29, 485, 2, 1722, 5446, 3, 8000, 10, 5300, 270, 181188, 955290, 4, 4, 15988040, 416012, 32420068, 2682744, 223, 25851, 8409205, 49871, 301, 1713301109422, 1066033105795, 4270, 57425882, 859704866, 11125766, 77746116, 39343318862281, 501010332520, 4762
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OFFSET
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2,4
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COMMENTS
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The length of row n is A008615(n), n >= 2.
The denominator triangle is given in A274343.
The Eisenstein series with even index, G_{2*n}, when multiplied by 2*n-1, namely c(n) := (2*n-1)*G_{2*n}, satisfy the well-known recurrence relation (n-3) * (2*n +1) * c(n) = 3 * Sum_{j=2..n-2} c(j) * c(n-j), for n >= 4, with initial terms c(2) = c2 and c(3) = c3. See, e.g., the references Abramowitz-Stegun, 18.5.3, p. 635, Apostol p. 13, and Tricomi, p. 34.
The solution of this recurrence is c(n) = Sum a(n, m)/A274343(n, m)*c2^e2(n, m)*c3^e3(n, m), where the sum is over the partitions of n with parts 2 and 3 only, and with nonnegative exponents e2(n, m) and e3(n, m), where m = 1..A008615(n). The order is by increasing number of parts. E.g., n=6 with the partitions 3^2 and 2^3, with c(6) = (1/13)*c(3)^2 + (2/39)*c(2)^3. See also the Abramowitz-Stegun reference 18.5.9 - 18.5.24, p. 636, for n=4..19, but not given in lowest terms, and with decreasing number of parts for the partitions (contrary to the listing of partitions on p. 831).
The rational numbers c(n) appear also as coefficients in the Laurent series of Weierstrass's P function: WeierstrassP(z; g_2, g_3) = 1/z^2 + Sum_{n >= 2} c(n) * z^{2*n-2}, with g_2 = 20*c(2) and g_4 = 28*c(3). See, e.g., the Abramowitz-Stegun reference 18.5.1, p. 635. See also the o.g.f. given below.
For the polynomials c(2)..c(20) see the W. Lang link, also for the corresponding Eisenstein series G_{2*n} in terms of g_2 and g_4.
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REFERENCES
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T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, p. 13.
F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948, pp. 34-35.
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LINKS
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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], ch. 18.5, pp. 635-636.
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FORMULA
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a(n) = numerator(r(n)) with the rationals r(n) in lowest terms obtained from the c(n) recurrence given in a comment above as coefficients of powers of c2 and c3 corresponding to the partitions of n with parts 2 and 3 only, when sorted with increasing number of parts.
O.g.f: C(x) = Sum_{n >= 2} c(n)*x^n = x*WeierstrassP(sqrt(x), g_2 = 20*c(2), g_3 = 28*c(3)) - 1. Compare with Abramowitz-Stegun, 18.5.1, p. 635.
Nonlinear differential equation of second order for the o.g.f C(x) derived from the recurrence relation of c(n): 2*x^2*(d^2/dx^2)C(x) - 3*x*(d/dx)C(x) - 3*C(x) + 5*x^2*c(2) - 3*C(x)^2 = 0, with C(0) = 0 and C'(0) = 0.
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EXAMPLE
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The irregular triangle a(n, m) begins:
n\m 1 2 3 ...
2: 1
3: 1
4: 1
5: 3
6: 1 2
7: 2
8: 60 5
9: 1 29
10: 485 2
11: 1722 5446
12: 3 8000 10
13: 5300 270
14: 181188 955290 4
15: 4 15988040 416012
16: 32420068 2682744 223
17: 25851 8409205 49871
...
row n = 18: 301 1713301109422 1066033105795 4270,
row n = 19: 57425882 859704866 11125766,
row n = 20: 77746116 39343318862281 501010332520 4762.
The irregular triangle of rationals r(n) starts:
n\m: 1 2 3 ...
2: 1/1
3: 1/1
4: 1/3
5: 3/11
6: 1/13 2/39
7: 2/33
8: 60/2431 5/663
9: 1/2 29/2717
10: 485/80223 2/1989
11: 1722/1062347 5446/3187041
12: 3/16055 8000/6605027 10/77571
13: 5300/11685817 270/1062347
...
row n = 14: 181188/2002524095 955290/4405553009 4/249951,
row n = 15: 4/497705 15988040/155409680283 416012/11559397707,
row n = 16: 32420068/1123416017295 2682744/74894401153 223/114727509,
row n = 17: 25851/5643476995 8409205/409716429837 49871/10158258591,
row n = 18: 301/909705199 1713301109422/233400836858808047 1066033105795/190964321066297493 4270/18394643943,
row n = 19: 57425882/34825896536145 859704866/229850917138557 11125766/17096349208653,
row n = 20: 77746116/357856262339147 39343318862281/24291640943843637507 501010332520/602272089516784401 4762/174041631153.
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CROSSREFS
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KEYWORD
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nonn,tabf,frac,easy
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AUTHOR
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STATUS
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approved
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