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 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A322194 E.g.f.: S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), where S(x,y) = Sum_{n>=0} Sum_{k=0..2*n+1} T(n,k) * x^(2*n+1-k)*y^k/((2*n+1-k)!*k!), as a triangle of coefficients T(n,k) read by rows. 4
 1, 1, 1, 2, 2, 1, 1, 8, 14, 14, 8, 1, 1, 32, 122, 200, 200, 122, 32, 1, 1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1, 1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1, 1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1, 1, 8192, 797162, 12584960, 73330760, 226744832, 446442272, 614111360, 614111360, 446442272, 226744832, 73330760, 12584960, 797162, 8192, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS See A322622 for another description of the e.g.f. of this sequence. LINKS Paul D. Hanna, Table of n, a(n) for n = 1..2652 FORMULA E.g.f.: S(x,y) and related series C(x,y) satisfy the following identities. (1) C(x,y)^2 - S(x,y)^2 = 1. (2a) C(x,y) = cosh(x) * cosh(y) / (1 - sinh(x)*sinh(y)). (2b) S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)). (3a) cosh(x) = C(x,y) * cosh(y) / (1 + sinh(y)*S(x,y)). (3b) sinh(x) = (S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)). (3c) cosh(y) = C(x,y) * cosh(x) / (1 + sinh(x)*S(x,y)). (3d) sinh(y) = (S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)). (4a) exp(x) = (C(x,y)*cosh(y) + S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)). (4b) exp(y) = (C(x,y)*cosh(x) + S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)). (5a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) * (cosh(y) - sinh(y)*C(x,y)) / (1 - sinh(y)^2*S(x,y)^2). (5b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) * (cosh(x) - sinh(x)*C(x,y)) / (1 - sinh(x)^2*S(x,y)^2). (5c) C(x,y) + S(x,y) = (cosh(x) + sinh(x)*cosh(y)) * (cosh(y) + sinh(y)*cosh(x)) / (1 - sinh(x)^2*sinh(y)^2). (6a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) / (cosh(y) + sinh(y)*C(x,y)). (6b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) / (cosh(x) + sinh(x)*C(x,y)). (6c) C(x,y) + S(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)). (6d) C(x,y) + S(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)). SPECIAL ARGUMENTS. S(x, y=0) = sinh(x). S(x, y=x) = 2*sinh(x) / (1 - sinh(x)^2). S(x, y=-x) = 0. EXAMPLE E.g.f.: S(x,y) = (1*x + 1*y) + (1*x^3/3! + 2*x^2*y/2! + 2*x*y^2/2! + 1*y^3/3!) + (1*x^5/5! + 8*x^4*y/4! + 14*x^3*y^2/(3!*2!) + 14*x^2*y^3/(2!*3!) + 8*x*y^4/4! + 1*y^5/5!) + (1*x^7/7! + 32*x^6*y/6! + 122*x^5*y^2/(5!*2!) + 200*x^4*y^3/(4!*3!) + 200*x^3*y^4/(3!*4!) + 122*x^2*y^5/(2!*5!) + 32*x*y^6/6! + 1*y^7/7!) + ... where S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)). This irregular triangle of coefficients of x^(2*n+1-k)*y^k/((2*n+1-k)!*k!) in C(x,y) begins 1, 1; 1, 2, 2, 1; 1, 8, 14, 14, 8, 1; 1, 32, 122, 200, 200, 122, 32, 1; 1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1; 1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1; 1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1; 1, 8192, 797162, 12584960, 73330760, 226744832, 446442272, 614111360, 614111360, 446442272, 226744832, 73330760, 12584960, 797162, 8192, 1; ... RELATED SERIES. The series C(x,y), such that C(x,y)^2 - S(x,y)^2 = 1, begins C(x,y) = 1 + (1*x^2/2! + 1*x*y + 1*y^2/2!) + (1*x^4/4! + 4*x^3*y/3! + 5*x^2*y^2/(2!*2!) + 4*x*y^3/3! + 1*y^4/4!) + (1*x^6/6! + 16*x^5*y/5! + 41*x^4*y^2/(4!*2!) + 52*x^3*y^3/(3!*3!) + 41*x^2*y^4/(2!*4!) + 16*x*y^5/5! + 1*y^6/6!) + (1*x^8/8! + 64*x^7*y/7! + 365*x^6*y^2/(6!*2!) + 784*x^5*y^3/(5!*3!) + 977*x^4*y^4/(4!*4!) + 784*x^3*y^5/(3!*5!) + 365*x^2*y^6/(2!*6!) + 64*x*y^7/7! + 1*y^8/8!) + ... The e.g.f. may be written with coefficients of x^(2*n+1-k)*y^k/(2*n+1)!, as follows: S(x,y) = (1*x + 1*y) + (1*x^3 + 6*x^2*y + 6*x*y^2 + 1*y^3)/3! + (1*x^5 + 40*x^4*y + 140*x^3*y^2 + 140*x^2*y^3 + 40*x*y^4 + 1*y^5)/5! + (1*x^7 + 224*x^6*y + 2562*x^5*y^2 + 7000*x^4*y^3 + 7000*x^3*y^4 + 2562*x^2*y^5 + 224*x*y^6 + 1*y^7)/7! + ... these coefficients are described by triangle A322622. MATHEMATICA T[n_, k_] := (2n-k+1)! k! SeriesCoefficient[(Sinh[x] + Sinh[y])/(1 - Sinh[x] Sinh[y]), {x, 0, 2n-k+1}, {y, 0, k}]; Table[T[n, k], {n, 0, 7}, {k, 0, 2n+1}] // Flatten (* Jean-François Alcover, Dec 29 2018 *) PROG (PARI) {T(n, k) = my(X=x+x*O(x^(2*n+1-k)), Y=y+y*O(y^k)); S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y)); (2*n+1-k)!*k!*polcoeff(polcoeff(S, 2*n+1-k, x), k, y)} /* Print as a triangle */ for(n=0, 10, for(k=0, 2*n+1, print1( T(n, k), ", ")); print("")) CROSSREFS Cf. A322190 (C + S), A322193 (C), A322196 (main diagonal). Cf. A322622. Sequence in context: A260360 A011296 A176602 * A174120 A240939 A016739 Adjacent sequences: A322191 A322192 A322193 * A322195 A322196 A322197 KEYWORD nonn,tabf AUTHOR Paul D. Hanna, Dec 20 2018 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 November 29 18:13 EST 2022. Contains 358431 sequences. (Running on oeis4.)