The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A352414 G.f. A(x) satisfies: A(x)^5 = (1-x) * (A(x) + x)^4. 2
 1, 3, -10, 50, -345, 2681, -22416, 196700, -1786715, 16656155, -158443468, 1531830328, -15007700345, 148672680185, -1486712621330, 14987306377954, -152144993493979, 1554005064929735, -15958686622754240, 164676857033422880 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Table of n, a(n) for n=0..19. FORMULA G.f. A(x) satisfies: (1) A(x) = ( A(x)^5/(1-x) )^(1/4) - x. (2) A(x)^5 = (1-x) * (A(x) + x)^4. (3) A( x*(1+x)^4/(1 + x*(1+x)^4) ) = (1+x)^4/(1 + x*(1+x)^4). (4) A(x) = x / Series_Reversion( x*(1+x)^4/(1 + x*(1+x)^4) ). (5) Sum_{k=0..n} [x^k] A(x)^n = (-1)^(n-1) * 4, for n >= 1. EXAMPLE G.f.: A(x) = 1 + 3*x - 10*x^2 + 50*x^3 - 345*x^4 + 2681*x^5 - 22416*x^6 + 196700*x^7 - 1786715*x^8 + ... where A(x)^5 equals (1-x)*(A(x) + x)^4, as can be seen from the following power series expansions: A(x)^5 = 1 + 15*x + 40*x^2 - 80*x^3 - 20*x^4 + 48*x^5 - 420*x^6 + 8160*x^7 - 109230*x^8 + ... (A(x) + x)^4 = 1 + 16*x + 56*x^2 - 24*x^3 - 44*x^4 + 4*x^5 - 416*x^6 + 7744*x^7 - 101486*x^8 + ... Related table. Another defining property of the g.f. A(x) is illustrated here. The table of coefficients of x^k in A(x)^n begins: n=1: [1, 3, -10, 50, -345, 2681, -22416, 196700, ...]; n=2: [1, 6, -11, 40, -290, 2292, -19346, 170784, ...]; n=3: [1, 9, -3, -3, -105, 1083, -10105, 94239, ...]; n=4: [1, 12, 14, -52, 21, 224, -3208, 35792, ...]; n=5: [1, 15, 40, -80, -20, 48, -420, 8160, ...]; n=6: [1, 18, 75, -60, -255, 294, -77, 720, ...]; n=7: [1, 21, 119, 35, -630, 350, 322, -214, ...]; ... in which the partial sum of row n up to column n equals (-1)^(n-1)*4, as illustrated by: n=1: 4 = 1 + 3; n=2: -4 = 1 + 6 + -11; n=3: 4 = 1 + 9 + -3 + -3; n=4: -4 = 1 + 12 + 14 + -52 + 21; n=5: 4 = 1 + 15 + 40 + -80 + -20 + 48; n=6: -4 = 1 + 18 + 75 + -60 + -255 + 294 + -77; n=7: 4 = 1 + 21 + 119 + 35 + -630 + 350 + 322 + -214; ... PROG (PARI) {a(n) = polcoeff( x/serreverse( x*(1+x)^4/(1 + x*(1+x)^4 +x^2*O(x^n)) ), n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A352385, A352413, A352415. Sequence in context: A209902 A049370 A009343 * A341648 A307099 A048175 Adjacent sequences: A352411 A352412 A352413 * A352415 A352416 A352417 KEYWORD sign AUTHOR Paul D. Hanna, Mar 15 2022 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 June 14 15:00 EDT 2024. Contains 373400 sequences. (Running on oeis4.)