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A274342 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.

%I A274342 #15 May 08 2018 15:11:56
%S A274342 1,1,1,3,1,2,2,60,5,1,29,485,2,1722,5446,3,8000,10,5300,270,181188,
%T A274342 955290,4,4,15988040,416012,32420068,2682744,223,25851,8409205,49871,
%U A274342 301,1713301109422,1066033105795,4270,57425882,859704866,11125766,77746116,39343318862281,501010332520,4762
%N A274342 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.
%C A274342 The length of row n is A008615(n), n >= 2.
%C A274342 The denominator triangle is given in A274343.
%C A274342 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.
%C A274342 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).
%C A274342 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.
%C A274342 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.
%D A274342 T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, p. 13.
%D A274342 F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948, pp. 34-35.
%H A274342 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], ch. 18.5, pp. 635-636.
%H A274342 Wolfdieter Lang, <a href="/A274342/a274342.pdf">Rationals c(n), n = 2..20, and Eisenstein series G_{2*k}, k = 2..10.</a>
%F A274342 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.
%F A274342 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.
%F A274342 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.
%e A274342 The irregular triangle a(n, m) begins:
%e A274342 n\m          1          2         3   ...
%e A274342 2:           1
%e A274342 3:           1
%e A274342 4:           1
%e A274342 5:           3
%e A274342 6:           1          2
%e A274342 7:           2
%e A274342 8:          60          5
%e A274342 9:           1         29
%e A274342 10:        485          2
%e A274342 11:       1722       5446
%e A274342 12:          3       8000        10
%e A274342 13:       5300        270
%e A274342 14:     181188     955290         4
%e A274342 15:          4   15988040    416012
%e A274342 16:   32420068    2682744       223
%e A274342 17:      25851    8409205     49871
%e A274342 ...
%e A274342 row n = 18: 301  1713301109422 1066033105795 4270,
%e A274342 row n = 19: 57425882 859704866 11125766,
%e A274342 row n = 20: 77746116 39343318862281 501010332520  4762.
%e A274342 The irregular triangle of rationals r(n) starts:
%e A274342 n\m:      1              2            3  ...
%e A274342 2:       1/1
%e A274342 3:       1/1
%e A274342 4:       1/3
%e A274342 5:       3/11
%e A274342 6:       1/13           2/39
%e A274342 7:       2/33
%e A274342 8:      60/2431         5/663
%e A274342 9:       1/2           29/2717
%e A274342 10:    485/80223        2/1989
%e A274342 11:   1722/1062347   5446/3187041
%e A274342 12:     3/16055      8000/6605027   10/77571
%e A274342 13:  5300/11685817   270/1062347
%e A274342 ...
%e A274342 row n = 14: 181188/2002524095 955290/4405553009  4/249951,
%e A274342 row n = 15: 4/497705  15988040/155409680283 416012/11559397707,
%e A274342 row n = 16: 32420068/1123416017295 2682744/74894401153  223/114727509,
%e A274342 row n = 17:  25851/5643476995    8409205/409716429837 49871/10158258591,
%e A274342 row n = 18: 301/909705199  1713301109422/233400836858808047  1066033105795/190964321066297493  4270/18394643943,
%e A274342 row n = 19: 57425882/34825896536145  859704866/229850917138557  11125766/17096349208653,
%e A274342 row n = 20: 77746116/357856262339147  39343318862281/24291640943843637507  501010332520/602272089516784401  4762/174041631153.
%Y A274342 Cf. A008615, A274343.
%K A274342 nonn,tabf,frac,easy
%O A274342 2,4
%A A274342 _Wolfdieter Lang_, Jun 20 2016