The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A326602 G.f.: Sum_{n>=0} (x^(2*n-1) + 1)^n * x^n / (1 + x^(2*n+1))^(n+1), an even function. 3
 1, 3, 0, 2, 9, -12, 8, 12, -10, 15, -36, 46, -18, 20, 12, -82, 101, -162, 302, -168, -100, 92, 32, 40, -244, 351, -452, 1052, -1528, 1144, -394, -494, 948, -438, 370, -1474, 2805, -2860, 2560, -4762, 6554, -4104, 926, 480, -1820, 3074, -2546, 1072, -2518, 9745, -17810, 21300, -28982, 37560, -26380, 6162, 686, 2, 2364, -12342, 30356, -39584, 19448, 7562, 9491, -63824, 99128, -116668 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..4100 FORMULA G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n), where (1) A(x) = Sum_{n>=0} (x^(2*n-1) + 1)^n * x^n / (1 + x^(2*n+1))^(n+1), (2) A(x) = Sum_{n>=0} (x^(2*n-1) - 1)^n * x^n / (1 - x^(2*n+1))^(n+1), (3) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * (x^(2*n-1) - x^(2*k))^(n-k), (4) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * (x^(2*n-1) + x^(2*k))^(n-k) * (-1)^k. a(n) is odd iff n is square (conjecture). EXAMPLE G.f.: A(x) = 1 + 3*x^2 + 2*x^6 + 9*x^8 - 12*x^10 + 8*x^12 + 12*x^14 - 10*x^16 + 15*x^18 - 36*x^20 + 46*x^22 - 18*x^24 + 20*x^26 + 12*x^28 - 82*x^30 + ... such that A(x) = 1/(1 + x) + (1 + x)*x/(1 + x^3)^2 + (1 + x^3)^2*x^2/(1 + x^5)^3 + (1 + x^5)^3*x^3/(1 + x^7)^4 + (1 + x^7)^4*x^4/(1 + x^9)^5 + (1 + x^9)^5*x^5/(1 + x^11)^6 + (1 + x^11)^6*x^6/(1 + x^13)^7 + (1 + x^13)^7*x^7/(1 + x^15)^8 + ... also A(x) = 1/(1 - x) - (1 - x)*x/(1 - x^3)^2 + (1 - x^3)^2*x^2/(1 - x^5)^3 - (1 - x^5)^3*x^3/(1 - x^7)^4 + (1 - x^7)^4*x^4/(1 - x^9)^5 - (1 - x^9)^5*x^5/(1 - x^11)^6 + (1 - x^11)^6*x^6/(1 - x^13)^7 - (1 - x^13)^7*x^7/(1 - x^15)^8 + ... AS A TRIANGLE. This sequence may be written as a triangle like so 1, 3, 0, 2, 9, -12, 8, 12, -10, 15, -36, 46, -18, 20, 12, -82, 101, -162, 302, -168, -100, 92, 32, 40, -244, 351, -452, 1052, -1528, 1144, -394, -494, 948, -438, 370, -1474, 2805, -2860, 2560, -4762, 6554, -4104, 926, 480, -1820, 3074, -2546, 1072, -2518, 9745, -17810, 21300, -28982, 37560, -26380, 6162, 686, 2, 2364, -12342, 30356, -39584, 19448, 7562, 9491, -63824, 99128, -116668, 167616, -212884, 156266, -32564, -35108, 21732, 3952, -4058, -19496, 89988, -200198, 251662, -198964, ... in which the terms a(n^2) form the leftmost column. The odd terms seem to occur only at a(n^2) and begin: [1, 3, 9, 15, 101, 351, 2805, 9745, 9491, 138675, 776675, 7430517, 43105515, ...]. PROG (PARI) {a(n) = my(A = sum(m=0, 2*n+1, (x^(2*m-1) - 1 +O(x^(2*n+2)) )^m * x^m / (1 - x^(2*m+1) +O(x^(2*n+2)) )^(m+1) )); polcoeff(A, 2*n)} for(n=0, 100, print1(a(n), ", ")) (PARI) {a(n) = my(A = sum(m=0, 2*n+1, (x^(2*m-1) + 1 +O(x^(2*n+2)) )^m * x^m / (1 + x^(2*m+1) +O(x^(2*n+2)) )^(m+1) )); polcoeff(A, 2*n)} for(n=0, 100, print1(a(n), ", ")) CROSSREFS Cf. A326603, A323557. Sequence in context: A231101 A193084 A126598 * A256548 A239098 A319501 Adjacent sequences: A326599 A326600 A326601 * A326603 A326604 A326605 KEYWORD sign AUTHOR Paul D. Hanna, Aug 08 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 19:03 EST 2023. Contains 367593 sequences. (Running on oeis4.)