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List of Riordan arrays
From OeisWiki
(from search string "Riordan keyword:tabl", the last 82 pages)
See also the article by Emeric Deutsch, Luca Ferrari and Simone Rinaldi, Production matrices and Riordan arrays
Contents
- 1 How to expand Riordan arrays
- 2 Ordinary RA
- 3 Inverses of rational R
- 4 Exponential Riordan arrays
- 5 Multidimensional tables
How to expand Riordan arrays
In Maple (cf. Question on ask.sagemath.org):
Riordan := (d,h,n,k) -> coeftayl(d*h^k,x=0,n)*n!/k!: A131222 := (n,k) -> Riordan(1,log((1-x)/(1-2*x)),n,k): seq(print(seq(A131222(n,k),k=0..n)),n=0..5);
Ordinary RA
Lagrange subgroup R(1,h(x))
- A099089 (1, 2+x)
- A099096 (1, 2-x)
- A099097 (1, 3+x)
- A099040 (1, 2+2x)
- A099095 (1, 3+2x)
- A099091 (1, 2+3x)
- A099092 (1, 2+4x)
- A099093 (1, 3+3x)
- A109466 (1, x(1-x))
- A109970 (1, x(1-x)^2)
- A026729 (1, x(1+x))
- A116088 (1, x(1+x)^2)
- A116089 (1, x(1+x)^3)
- A113953 (1, x(1+2x))
- A110509 (1, x(1-2x))
- A110517 (1, x(1-3x))
- A124304 (1, x(1-x^2))
- A071675 (1, x(1+x+x^2))
- A112517 (1, x(1-2x^2-x^3))
- A213887 (1, x+x^2+x^3+x^4)
- A097805 (1, 1/(1-x))
- A122542 (1, x(1+x)/(1-x))
- A122016 (1, x(1+2x)/(1-x))
- A207327 (1, x(1+x)^2/(1-x))
- A105477 (1, x(1+x-x^2)/(1-x))
- A194582 (1, x(1+2x-x^2)/(1-x))
- A128908 (1, x/(1-x)^2)
- A206294 (1, x/(1-x)^3)
- A060086 (1, x/((1+x)*(1-x)^2))
- A198295 (1, x(1+x)/(1-x^3))
- A062110 (1, x(1-x)/(1-2x))
- A147720 (1, x(1-x)/(1-3x))
- A155161 (1, x/(1-x-x^2))
- A155112 (1, x(1+x)/(1-x-x^2))
- A204040 (1, x(1+x)/(1-x+2x^2))
- A188137 (1, x(1-x)/(1-3x+x^2))
- A204533 (1, x(1-x)^2/(1-3x+x^2))
- A147746 (1, x(1-2x)/(1-3x+x^2))
- A206306 (1, x/(1-3x+2x^2))
- A147723 (1, x(1-4x)/(1-7x+3x^2))
- A084938 (1, x*A000142(x))
- A111106 (1, x*A001147(x))
- A122538 (1, x*A006318(x))
- A167749 (1, x*A005169(x))
- A111146, A205574
Appell subgroup R(d(x),x)
- A185018 ((1-x)^2, x)
- A167194 ((1+x)^2, x)
- A122431 ((1+x)^3, x)
- A114284 ((1-3x)/(1-x), x)
- A113680 ((1-x-2x^2)/(1-x) ,x)
- A101038 ((1-x)^2/(1+x), x)
- A025581 (x/(1-x)^2, x)
- A099375 ((1+x)/(1-x)^2, x)
- A122432 (1/(1+x)^3, x)
- A157038 ((1-x)/(1-2x), x)
- A114283 ((1-x)/(1-3x), x)
- A128174 (1/(1-x^2), x)
- A117568 ((1-x^3)/(1+x^2), x)
- A116949 ((1-x^3)/(1+2x^2), x)
- A104762 (1/(1-x-x^2), x)
- A113678 ((1-x)/(1-x-2x^2), x)
- A128514 (1/(1-2x-x^2), x)
- A111567 ((1+x^2)/(1-x^3), x)
- A116948 ((1+2x^2)/(1-x^3), x)
- A104975 (x^2/(sum{k>=0, x^(2^k)}-x), x)
- A106476
Bell subgroup R(d(x),xd(x)) (convolution triangles)
- A030528 (1+x, x(1+x))
- A007318 (1/(1-x), x/(1-x))
- A113413 ((1+x)/(1-x), x(1+x)/(1-x))
- A078812 (1/(1-x)^2, x/(1-x)^2)
- A208904 ((1+x)/(1-x)^2, x(1+x)/(1-x)^2)
- A124394 ((1-2x)/(1-x)^2, x(1-2x)/(1-x)^2)
- A124819 (1/(1-x)^2, x/(1-x)^3)
- A127893 (1/(1-x)^3, x/(1-x)^3)
- A130595 (1/(1+x), x/(1+x))
- A053122 (1/(1+x)^2, x/(1+x)^2)
- A127895 (1/(1+x)^3, x/(1+x)^3)
- A105495 ((1+x)/(1-x)^3, x(1+x)/(1-x)^3)
- A168561 (1/(1-x^2), x/(1-x^2))
- A105475 ((1+2x)/(1-x^2),x(1+2x)/(1-x^2))
- A038207 (1/(1-2x), x/(1-2x))
- A105306 ((1-x)/(1-2x), (x-x^2)/(1-2x))
- A116412 ((1+x)/(1-2x), x(1+x)/(1-2x))
- A125693 ((1-x)/(1-3x), x(1-x)/(1-3x))
- A038231 (1/(1-4x), x/(1-4x))
- A110441 (1/((1-x)(1-2x)), x/((1-x)(1-2x)))
- A116414 (1/((1-x)(1-3x)), x/((1-x)(1-3x)))
- A136688 (1/(1-2x^2), x/(1-2x^2))
- A037027 (1/(1-x-x^2), x/(1-x-x^2))
- A154929 ((1+x)/(1-x-x^2), x(1+x)/(1-x-x^2))
- A101950 (1/(1-x+x^2), x/(1-x+x^2))
- A185962 ((1-x)^2/(1-x+x^2), x(1-x)^2/(1-x+x^2))
- A104562 (1/(1+x+x^2), x/(1+x+x^2))
- A073370 (1/(1-x-2x^2), x/(1-x-2x^2))
- A054456 (1/(1-2x-x^2), x/(1-2x-x^2))
- A104597 ((1-x)/(1-x+x^2), x(1-x)/(1-x+x^2))
- A125662 (1/(1-3x+x^2), x/(1-3x+x^2))
- A207823 (1/(1-4x+x^2), x/(1-4x+x^2))
- A207824 (1/(1-5x+x^2), x/(1-5x+x^2))
- A207815 (1/(1+3x+x^2), x/(1+3x+x^2))
- A112899 (1/(1-2x-x^2), x/(1-2x-x^2))
- A054458 ((1+x)/(1-2*x-x^2), (x+x^2)/(1-2x-x^2))
- A190215 ((1-x-x^2)/(1-2x-x^2), (x-x^2-x^3)/(1-2x-x^2))
- A132964 (1/(1-3x-x^2), x/(1-3x-x^2)
- A117316 ((1-x)/(1-x-2x^2), x(1-x)/(1-x-2x^2))
- A159764 (1/(1+4x+x^2), x/(1+4x+x^2))
- A123967 (1/(1+5x+x^2), x/(1+5x+x^2))
- A111806 (1/(1+3x+2x^2),x/(1+3x+2x^2))
- A206819 (1/(1-10x-10x^2), x/(1-10x-10x^2))
- A051594 (1/((1-x^2)*(1-x)), x/((1-x^2)*(1-x)))
- A126030 (1/(1+x^3), x/(1+x^3))
- A104578 (1/(1-x^2-x^3), x/(1-x^2-x^3))
- A104580 (1/(1-x-x^2-x^3), x/(1-x-x^2-x^3)
- A208153 (1/(1-x-2x^2+x^3), x/(1-x-2x^2+x^3)
- A104579 (1/(1-x^2-2x^3), x(1-x^2-2x^3)
- A131321 ((1-x^2)/(1-3*x^2+x^4), x(1-x^2)/(1-3*x^2+x^4))
- A065600 (A000957(x), x*A000957(x))
- A092726 (A006013(x), x*A006013(x))
- A171224 (A117641(x), x*A117641(x))
- A171486 (A033321(x), x*A033321(x))
- A171488 (A005773(x+1),x*A005773(x+1))
- A171505 (A059738(x), x*A059738(x))
- A171509 (A126931(x), x*A126931(x))
- A171515 (A033543(x), x*A033543(x))
- A171567 (A168491(x), x*A168491(x))
- A171568 (A064613(x), x*A064613(x))
- A171589 (A104455(x), x*A104455(x))
- A172026 (A001764(x^2), x*A001764(x^2))
- A202847 (A126930(x), x*A126930(x))
Hitting time arrays R(xh'(x)/h(x),xh(x))
More Ordinary RA (proper or improper)
h(x) not a fraction
- A167374 (1-x, 1)
- A097806 (1+x, 1)
- A097807 (1/(1+x), 1)
- A110515 ((1-x+x^2+x^3)/(1-x^4), 1)
- A100218 ((1-2x)/(1-x), 1-x)
- A098599 ((1+2x)/(1+x), 1+x)
- A117362 (1-2x, x(1-x))
- A113214 (1+2x, x(1+x))
- A117372 (1-3x, x(1-x))
- A117377 (1-4x, x(1-x))
- A109264 (1-x-x^2, x(1-x))
- A109246 (1-x-2x^2, x(1-x))
- A119305 (1-4x, x(1-x)^3)
- A167431 (1-4x+4x^2, x(1-2x))
- A004070 (1/(1-x), x(1+x))
- A104726 (1/(1-x-x^2), x(1+x))
- A124234 (1/(1-x), x(1+x)^2)
- A119302 (1-3x, x(1-x)^2)
- A110291 (1/(1-x), x(1+2x))
- A122438 (1/(1-2x), x(1+2x))
- A214582 (1/(1-x-x^2), x*(1+2x))
- A107065 (1/(1-x), x(1+x+x^2+x^3))
- A191238 (1+x^2+x^4, x+x^3+x^5)
- A191239 (1+x+2x^3, x+x^2+2x^3)
h(x) denominator a power of 1-x
- A159854 (1-x, x/(1-x))
- A131084 (1+x, x/(1-x))
- A113313 (1-2x, x/(1-x))
- A202241 ((1+x)^2, x/(1-x))
- A104402 (1-x+x^2, x(1-x))
- A052553 (1/(1-x), x^2/(1-x))
- A029653 ((1+x)/(1-x), x/(1-x))
- A029635 ((2-x)/(1-x), x/(1-x))
- A112467 ((1-2x)/(1-x), x/(1-x))
- A093560 ((1+2x)/(1-x), 1/(1-x))
- A095660 ((3-2x)/(1-x), 1/(1-x))
- A072405 ((1-x+x^2)/(1-x), x/(1-x))
- A159855 ((1-2x-x^2)/(1-x), x/(1-x))
- A159853 ((1-2x+2x^2)/(1-x), x/(1-x))
- A135278 (1/(1-x)^2, x/(1-x))
- A104712 (1/(1-x)^3, x/(1-x))
- A104713 (1/(1-x)^4, x/(1-x))
- A110813 ((1+x)/(1-x)^2, x/(1-x))
- A125165 ((1+x)/(1-x)^3, x/(1-x))
- A233295 ((1+x)/(1-x)^3, 2x/(1-x))
- A124725 ((1-2x+2x^2)/(1-x)^3, x/(1-x))
- A112465 (1/(1+x), x/(1-x))
- A055248 (1/(1-2x), x/(1-x))
- A106516 (1/(1-3x), x/(1-x))
- A128175 ((1-x)/(1-2x), x/(1-x))
- A055249 ((1-x)/(1-2x)^2, x/(1-x))
- A055252 ((1-x)^2/(1-2x)^3, x/(1-x))
- A055584 ((1-x)^3/(1-2x)^4, x/(1-x))
- A104709 (1/((1-x)(1-2x)), x/(1-x))
- A191582 (1/(1-3x^2), x/(1-x))
- A105809 (1/(1-x-x^2), x/(1-x))
- A106513 (1/(1-2x-x^2), x/(1-x))
- A091597 (1/(1-x-2x^2), x/(1-x))
- A099567 (1/(1-x-x^3), 1/(1-x))
- A125171 ((1-x)/(1-3x+x^2), x/(1-x))
- A106522 (1/(1-x-x^2-x^3), x/(1-x))
- A008288 (1/(1-x), x(1+x)/(1-x))
- A104698 (1/(1-x)^2, x(1+x)/(1-x))
- A112475 (1/(1+x), x(1+x)/(1-x))
- A101603 (1/(1-x^2), x(1+x)/(1-x))
- A154948 ((1+x)/(1-x^2)^2, x(1+x)/(1-x))
- A158687 (1/(1-x), x(1+x)^2/(1-x))
- A124279 (1/(1-x), x(1-x+x^2)/(1-x))
- A103141 (1/(1-x), x(1+x+x^2+x^3)/(1-x))
- A098593 (1/(1-x), (1-2x)/(1-x))
- A117352 (1/(1-2x), x(1-2x)/(1-x))
- A081577 (1/(1-x), x(1+2x)/(1-x))
- A123562 (1/(1-x), x(1-3x)/(1-x))
- A081578 (1/(1-x), x(1+3x)/(1-x))
- A085478 (1/(1-x), x/(1-x)^2)
- A123970 (1/(1-x), -x/(1-x)^2)
- A111125 ((1+x)/(1-x)^2, x/(1-x)^2)
- A136672 ((1+x^2)/(1-x)^2, -x/(1-x)^2)
- A158909 (1/((1-x)(1-x^2)), x/(1-x)^2)
- A131250 (1/(1-2x), x/(1-x)^2)
- A211955 (x/(1-x)^3, 2x/(1-x)^2)
- A135552 (1/((1-2x)(1-x)^2), -x/(1-x)^2)
- A202023 (1/(1-x), x^2/(1-x)^2)
- A202064 (x/(1-x)^2, x^2/(1-x)^2)
- A114188 (1/(1-x), x(1+x)/(1-x)^2)
- A114123 (1/(1-x), x(1+x)^2/(1-x)^2)
- A216182 ((1+x)/(1-x)^2, x(1+x)^2/(1-x)^2)
- A202480 (1/(1-x), x(2x-1)/(1-x)^2)
- A109955 (1/(1-x), x/(1-x)^3)
- A109960 (1/(1-x), x/(1-x)^4)
h(x) denominator a power of 1+x
- A122433 ((1+x)^2, x/(1+x))
- A112468 (1/(1-x), x/(1+x))
- A113310 ((1+x)/(1-x), x/(1+x))
- A059260 (1/(1-x^2), 1/(1+x))
- A112466 ((1+2x)/(1+x), x/(1+x))
- A097808 ((1+2x)/(1+x)^2, 1/(1+x))
- A105810 ((1+x-x^2)/(1+x)^2, x/(1+x))
- A106509 ((1+x)/(1+x+x^2), x/(1+x))
- A099569 ((1+x)^2-x^3)/(1+x)^3, 1/(1+x))
- A124377 (1/(1-x-x^2), x/(1+x))
- A103316 (1/(1+2x), x/(1+x))
- A110814 (1/(1+3x+2x^2), x/(1+x))
- A186827 (1-x, x(1-x)/(1+x))
- A129818 (1/(1+x), x/(1+x)^2)
- A053124 (1/(1+x)^2, 4x/(1+x)^2)
- A109954 (1/(1+x)^3, x/(1+x)^2)
- A110162 ((1-x)/(1+x), x/(1+x)^2)
- A122917 (1/(1+x+x^2),x/(1+x)^2)
- A158454 (1/(1-x^2), x/(1+x)^2)
- A110511 (1/(1+x), x(1-x)/(1+x)^2)
- A124341 (1/(1+x), x(1+2x)/(1+x)^2)
- A110522 (1/(1+x), x(1-2x)/(1+x)^2)
More rational h(x)
- A046854 (1/(1-x), x/(1-x^2))
- A152815 (1/(1-x), x^2/(1-x^2))
- A105438 (1/(1-x)^2, x/(1-x^2))
- A206474 ((1+x-x^2)/(1-x^2), x/(1-x^2))
- A111526 ((1+x^2)/(1-x^2)^2, x/(1-x^2))
- A030111 (1/((1-x)(1-x^2)), x/(1-x^2))
- A117355 (1/(1-x^2), x(1-2x^2)/(1-x^2))
- A112743 (1/(1-x^2), x(1+x^2)/(1-x^2))
- A060098 (1/(1-x), x/((1-x)*(1-x^2)))
- A188316 (1/(1-x^2), x/((1-x)*(1-x^2)))
- A130777 ((1-x)/(1+x^2), x/(1+x^2))
- A108045 ((1-x^2)/(1+x^2), x/(1+x^2))
- A128494 (1/((1+x^2)*(1-x)), x/(1+x^2))
- A112552 (1/(1+x^2)^2, x/(1+x^2))
- A165620 ((1-x)/(1-x^4), x/(1+x^2))
- A154950 (1/(1-x^4), x(1+x)/(1+x^2))
- A127672 ((1-x^2)/(1+x^2), x/(1+x^2))
- A185331 ((1-x+x^2)/(1+x^2), x/(1+x^2))
- A117179 ((1-x^2)/(1+x^2)^2, x/(1+x^2))
- A117185 ((1-x^2)/(1+x+2x^2+x^3+x^4), x/(1+x^2))
- A206831 ((1+x)/(1+x^2), x(1-x)/(1+x^2))
- A109223 (1/(1-x), x/(1-x^2)^2)
- A108756 ((1+x-x^2)/(1-x^2)^2, x/(1-x^2)^2)
- A156062 (1/(1-x^4), x/(1-x^4))
- A118801 (1/(1-x), -x/(1-2x))
- A065109 (1/(1-2x), -x/(1-2x))
- A200139 ((1-x)/(1-2x), x/(1-2x))
- A118800 ((1-x)/(1-2x), -x/(1-2x))
- A201701 ((1-x)/(1-2x), x^2/(1-2x))
- A201780 ((1-x)^2/(1-2x), x/(1-2x))
- A209149 ((1+x)/(1-2x), x/(1-2x))
- A112626 (1/(1-3x), x/(1-2x))
- A055587 (1/(1-x)), x(1-x)/(1-2x))
- A106195 (1/(1-2x), x(1-x)/(1-2x))
- A121574 (1/(1-2x), x(1+x)/(1-2x))
- A147750 ((1-x-x^2)/(1-x-2x^2), x(1-x)/(1-2x))
- A124237 (1/(1-2x), x/((1-x)(1-2x)))
- A193723 ((1-x)/(1-3x), x/(1-3x))
- A183189 ((1-x)/(1-3x), x^2/(1-3x))
- A193843 (1/((1-x)(1-3x)), x/(1-3x))
- A114192 (1/(1-2x), x/(1-2x)^2)
- A114164 (1/(1-2x), x(1-x)/(1-2x)^2)
- A128414 ((1-2x)/(1+2x), x/(1+2x)^2)
- A123876 (1/(1+2x), x(1+x)/(1+2x)^2)
- A114195 (1/(1-3x), x(1-x)/(1-3x)^2)
- A123486 (1/(1-2x), x/(1-4x^2))
- A091186 (1/(1-x), x/(1-x-x^2)
- A182001 ((2x+1)/(1-x-x^2), x/(1-x-x^2))
- A123265 ((2-x)/(1-x-x^2), x/(1-x-x^2))
- A152440 (1/(1-x-x^2), x/(1-x-x^2)^2)
- A053538 (1/(1-x-x^2), x(1-x)/(1-x-x^2))
- A063967 (1/(1-x-x^2), x(1+x)/(1-x-x^2))
- A112973 (1/(1-x-x^2), x(1+x)/(1-x-x^2)^2)
- A209745 (1/(1-x-x^2), x(2+x)/(1-x-x^2))
- A209746 ((1+x)/(1-x-x^2), x(2+x)/(1-x-x^2))
- A060922 ((1+2x)/(1-x-x^2), x(1+2x)/(1-x-x^2))
- A210636 ((1-x)/(1-2*x-x^2), x(1+x)/(1-2*x-x^2))
- A125690 (1/(1-x-2x^2), x(1-x)/(1-x-2x^2))
- A129267 (1/(1-x+x^2), x(1-x)/(1-x+x^2))
- A202551 (1/(1-x+x^2), -x(1-x)/(1-x+x^2))
- A102587 ((1-x^2)/(1+x+x^2), x/(1+x+x^2))
- A167655 (1-u,u) where u=x/(1+x+x^2)
- A202209 ((1-x)/(1-3x+x^2), x^2/(1-3x+x^2))
- A206800 (1/(1-3x+x^2), x(1-x)/(1-3x+x^2))
- A147703 ((1-2x)/(1-3x+x^2), x(1-x)/(1-3x+x^2))
- A110168 ((1-x^2)/(1+3x+x^2), x/(1+3x+x^2))
- A147721 ((1-3x)/(1-4x+x^2), x(1-x)/(1-4x+x^2))
- A201730 ((1-2x)/(1-4x+3x^2), x^2/(1-4x+3x^2))
- A147747 ((1-3x+x^2)/(1-4x+3x^2), x(1-2x)/(1-4x+3x^2))
- A181472 ((1+x)/(1+2x+2x^2), x(1+x)/(1+2x+2x^2))
- A147724 ((1-7x+3x^2)/(1-8x+7x^2), x(1-4x)/(1-8x+7x^2)
- A123878 ((1-x)/(1-x+x^2), x(1-x)^2/(1-x+x^2)^2)
- A124369 (1/((1-x-x^2)(1+x+x^2)), x(1+x)/((1-x-x^2)(1+x+x^2)))
- A189187 (1/(1-x-x^2-x^3), (x+x^2)/(1-x-x^2-x^3))
- A187889 (1/(1-x-x^2-x^3), (x+x^2+x^3)/(1-x-x^2-x^3)
- A125906,A123967,A103209,A054446
Inverses of rational R
In terms of sqrt() only
u = sqrt(1-4x)
- A205813 (1, x/u)
- A106187 (1/u, x)
- A106190 (u, x)
- A187888 (1/u, x/(1-x))
- A046521 (1/u, x/(1-4x))
- A155788 (1/(x+u), x/(x+u))
- A187887 (1/(u(1-x)), x/(1-x))
- A106270 (2u/(1+u), x)
- A187926 (1/(1-4x), (1-u)/2)
- A188481 (1/(1-4x), (1-u)/(2u))
- A054335 (1/u, x/u)
- A116395 (1/u, (1/u-1)/2)
- A189175 (1+x/u, (1-u)/2)
- A128899 (1, (1-2x-u)/(2x))
- A188513 (1/(x+u), (1-u)/(2(x+u))
- A094527 (1/u, (1-2x-u)/(2x))
- A039598 ((1-2x-u)/(2x^2),(1-2x-u)/(2x))
- A124392 ((1-u)/(xu(3-u)), (1-u)/(3-u))
- A200965 ((1-u)/(2x(1-x)),(1-u)/(2(1-x)))
- A122919 ((1-3x+2x^2-u(1-x))/(2x^2), (1-2x-u)/(2x))
u = sqrt(1-2x-3^2)
- A185812 (1/(1-x), 2x/(1+x+u))
- A185813 (x/(1-x-x^2), x(1+x-u)/(2x(1+x)))
- A111960 (1/u, x/u)
- A116392 (1/u, 1/u-1)
- A064189 ((1-x-u)/(2x^2), (1-x-u)/(2x))
- A127501 (1/u, (1+x-u)/(2u))
- A094531 (1/u, (1-x-u)/(2x))
- A185810 ((1+x-u)/(2x(1-x)),(1+x-u))
- A097609 ((1+x-u)/(2x(1-x)),(1+x-u)/(2(1-x)))
- A089942 ((1+x-u)/(2x(1+x)), (1-x-u)/(2x))
- A038622 ((-1+3x+u)/(2x(1-3x)), (1-x-u)/(2x))
- A115990 (1/u, ((1-2x-3x^2)/(2(1-3x))-u/2)
- A202710 (((1-x-2x^2-u)/(2x)),((1-x-2x^2-u)/(2x^2)))
u = sqrt(1-6x+x^2)
- A118384 (1/u, (1-3x-u)/(4x))
- A117394 (1-x+u)/2, (1+x-u)/4)
- A104209 ((1+x-u)/(4x), x(1+x-u)/(4x))
- A133367 ((1-x-u)/(2x), (1-3x-u)/(4x))
- A110440 (x(1-3x-u)/(4x), (1-3x-u)/(4x))
- A114709 ((1+3x-u)/(2x(2x+3)), (1+3x-u)/(2(2x+3)))
- A110171 ((1+x+u)/(2u),(1-x-u)/2)
Mixed or other poynomials in sqrt()
- A054336 ((sqrt(1+2x)-sqrt(1-2x))/(2x*sqrt(1-2x)), (sqrt(1+2x)-sqrt(1-2x))/(2sqrt(1-2x)))
- A141343 (1, x/sqrt(2-sqrt(1+4x))
- A113187 ((sqrt(1+4x)-1)/(2x*sqrt(1+4x)), (1+2x-sqrt(1+4x))/(2x))
- A128417 (1/sqrt(1-8x), (1-4x-sqrt(1-8x)/(8x))
- A110292 (1-u,u) where u=(-1+sqrt(1+8x))/4
- A122440 ((3-sqrt(1+8x))/2, (sqrt(1+8x)-1)/4)
- A111959 (1, x/sqrt(1-4*x^2))
- A201093 (1, x*A097331)
- A108044 (1/sqrt(1-4x^2), (1-sqrt(1-4x^2))/(2*x))
- A116389 (1/sqrt(1-4x^2), (1+x)/sqrt(1-4x^2)-1)
- A111963 (1/(sqrt(1+4x^2)+x), x/(sqrt(1+4x^2)+x))
- A106180 (1-y, y) where y=-(1-sqrt(1+4x^2))/(2x)
- A124448 (sqrt(1+4x^2)-2x, (1+2x-sqrt(1+4x^2))/2)
- A120616 (1/sqrt(1+4x^2), (1-sqrt(1+4x^2))/(2x))
- A105522 ((1+2x+4x^2-(1+2x)sqrt(1+4x^2))/(2x^2), (sqrt(1+4x^2)-1)/(2x))
- A160905 (1/sqrt((1+x-x^2)(1-3x-x^2)), (1-x-x^2-sqrt((1+x-x^2)(1-3x-x^2))/(2x))
- A080245 ((sqrt(1+6x+x^2)-x-1)/(2x), (sqrt(1+6x+x^2)-x-1)/2)
- A112477 ((1-x+sqrt(1+6x+x^2))/2, (sqrt(1+6x+x^2)-x-1)/2)
- A121576 (2-2x-sqrt(1-8x+4x^2), (1-2x-sqrt(1-8x+4x^2))/2)
- A121575 (-sqrt(4x^2+8x+1)+2x+2), (sqrt(4x^2+8x+1)-2x-1)/2)
- A104505 (1/sqrt(1-2x+5x^2), (1-x-sqrt(1-2x+5x^2))/(2x))
- A115991 (1/sqrt(1-2x-7x^2),(1+x-sqrt(1-2x-7x^2))/2)
- A200756 ((1-sqrt(1-4x-4x^2))/(2x),(1-sqrt(1-4x-4x^2))/2))
- A111965 (1/sqrt(1-6x+5x^2), x/(1-6x+5x^2))
- A091698 (1/(1+x), (sqrt(1+6x+5x^2)-x-1)/(2(1+x)))
- A110877 (2/(1+x+sqrt(1-6x+5x^2)),(1-3x-sqrt(1-6x+5x^2))/(2x))
- A091965 ((1-3x-sqrt(1-6x+5x^2))/(2x^2), (1-3x-sqrt(1-6x+5x^2))/(2x))
- A110165 (1/sqrt(1-6x+5x^2), (1-3x-sqrt(1-6x+5x^2))/(2x)
- A125694 ((1+3x-sqrt(1+2x+9x^2))/(2x), (1+3x-sqrt(1+2x+9x^2))/2)
- A052179 ((1-4x-sqrt(1-8x+12x^2))/(2x^2), (1-4x-sqrt(1-8x+12x^2))/(2x))
- A168151 (1/u,(1-u)/2), u=sqrt(1-4x+4*x^3)
- A132276 (g(x),xg(x)), where g(x) = (1-x-x^2-sqrt(1-2x-5x^2+2x^3+x^4))/(2x^2)
- A124305 (1, 2*sqrt(3)*sin(asin(3*sqrt(3)*x/2)/3)/3)
- A190252, A185967, A187913, A187914, A171243, A213221, A104259, A111373, A136018, A110438-9
Using c(x), M(x),...
- A106566 (1, xc(x))
- A033184 (c(x), xc(x))
- A122897 (1/(1-x), c(x)-1)
- A125177 (c(x), x/(1-x))
- A091491 (1/(1-x), xc(x))
- A185943 (1/(1-x)^2, xc(x))
- A185944 (1/(1-x)^3, xc(x))
- A185945 (1/(1-x)^4, xc(x))
- A168377 (1/(1+x), xc(x))
- A139375 (1/(1-x-x^2), xc(x))
- A185937 (x/(1-x-x^2), xc(x))
- A185675 ((1+x)/(1-x-x^2), xc(x))
- A185678 ((1+x)/(1-x-x^2))^3, xc(x))
- A185663 ((x/(1-x-x^2))^2, xc(x))
- A185663 ((x/(1-x-x^2))^3, xc(x))
- A185676 (((1+x)/(1-x-x^2))^2, xc(x))
- A139377 (1/(1-x-2x^2), xc(x))
- A112413 ((1-x)*c(x), x)
- A106268 (1/(2-c(x)), x)
- A100100 (1/(2-c(x)), xc(x))
- A167772 (c(x)/(1+xc(x)), xc(x))
- A117375 (1/(1-3xc(x)), xc(x))
- A117380 (1/(1-4xc(x)), xc(x))
- A168216 (1/(1-x), xc(x)/(1-xc(x)))
- A090299 (c(x), x/sqrt(1-4x))
- A092392 (1/sqrt(1-4x), xc(x))
- A035324 (1, c(x)/sqrt(1-4x))
- A039599 (c(x), xc(x)^2)
- A054445 (c(x)^2, xc(x))
- A090285 (c(x), xc(x)^2/(1-xc(x)^2))
- A109267 (1/(1-xc(x)-x^2*c(x)^2), xc(x))
- A109244 (1/(1-xc(x)-2x^2*c(x)^2), xc(x))
- A174302 ((1-x^2c(x)^2)/(1-xc(x)-x^2*c(x)^2), xc(x))
- A050155 (c(x)^3,xc(x)^2)
- A099039 (1, c(-x))
- A079513 (2/(2-xc(x)+xc(-x)), xc(x))
- A097761 (1-xc(-x))c(-x), xc(-x)^2)
- A159971 (2c(-x)-1, xc(-x)^3)
- A110510 (1, xc(2x))
- A167432 (c(2x)^2, xc(2x))
- A115193 ((1+2xc(2x))/(1+x), xc(2x))
- A110506 (1/(1-xc(2x)), xc(2x)/(1-xc(2x)))
- A114189 (1/(1+xc(-2x)), xc(-2x)/(1+xc(-2x))
- A114193 (1/(1+2xc(-2x)), xc(-2x)/(1+2xc(-2x))
- A110518 (1, xc(3x))
- A116866 ((1+3xc(3x)/2)/(1+x/2), xc(3x))
- A110519 (1/(1-xc(3x)), xc(3x)/(1-xc(3x)))
- A053121 (c(x^2), xc(x^2))
- A112554 (c(x^2)^2, xc(x^2))
- A165621 (c(x^2)*(1+xc(x^2)), xc(x^2))
- A126075 (c(x^2)/(1-2xc(x^2)),xc(x^2))
- A111418 (c(x)/sqrt(1-4x), xc(x)^2)
- A134388 ((1+2x)/sqrt(1-4x^2), xc(x^2))
- A117184 ((1+x)c(x^2)/sqrt(1-4x^2), xc(x^2))
- A116382 (1/sqrt(1-4x^2), (1-2x^2*c(x^2))(x^2*c(x^2))/(x(1-x-x^2*c(x^2)))
- A155761 (c(2x^2), xc(2x^2))
- A125692 (1-xc(-x^2), x(1-xc(-x^2))
- A111527 (c^2(-x^2)(1+x^2c(-x^2))/sqrt(1+4x^2), xc(-x^2))
- A117178 (c(x^2)/sqrt(1-4x^2),xc(x^2))
- A061554 (1/(1-x-x^2*c(x^2)), xc(x^2))
- A113408 (1/(1-x^2-x^4*c(x^4)),xc(x^2))
- A126953 (2/(1-6x+sqrt(1-4x^2)), xc(x^2))
- A127631 (1, xc(x)*c(xc(x)))
- A112519 (1, xc(x)*c(-xc(x)))
- A113955 (1/((1-4x)c(x)), xc(x)/sqrt(1-4x))
- A159965 (1/sqrt(1-4x), (1-2x-(1-3x)c(x))/(x*sqrt(1-4x)))
- A155887 (1, (1/(1-x))c(x/(1-x)))
- A154930 ((1/(1+x))c(-x/(1+x)), (x/(1+x))c(x/(1+x)))
- A122896 (1, xM(x))
- A167630 (1/(1-x), xM(x))
- A114422 (1/sqrt(1-2x-3x^2), M(x)-1)
- A172040 (1, 2xs(x))
- A172094 (s(x), s(x)-1)
- A186826 (s(x), xS(x))
- A103136 (1-xS(-x), xS(-x))
- A110271 ((1-xS(-x))^2, xS(-x))
- A187537,A065602,A164585
Inverses without known d(x),h(x)
- A010027,A109956, A109962, A156064, A127894, A127898, A188317, A188463, A146314, A181289, A071945, A127543, A158793, A158815, A181304, A119301, A124790, A181308, A181330, A181332, A181336, A103778, A104974, A119304, A181302, A161009
Exponential Riordan arrays
Lagrange subgroup
- A122848 (1, x(1+x/2))
- A187084 (1, 4x+6x^2+4x^3+x^4)
- A111596 (1, x/(1+x))
- A122850 (1, sqrt(1+2x)-1)
- A059297 (1, x*exp(x))
- A185690 (1, sin(x))
- A188832 (1, x/cos(x))
- A185951 (1, x*cosh(x))
- A111593 (1, tanh(x))
- A111594 (1, arctanh(x))
- A131222 (1, log((1-x)/(1-2x))
- A206022 (1, x*exp(asinh(-2x))
- A147311 (1, asin(tanh(x)))
- A147342 (1, log(sec(x)+tan(x)))
- A137452 (1, W(x))
Appell subgroup
- A129685 (1-x^2/2, x)
- A094587 (1/(1-x), x)
- A132159 (1/(1-x)^2, x)
- A129684 (1/(1-x^2/2), x)
- A136215 (1/(1-3x)^(1/3), x)
- A136216 (1/(1-3x)^(2/3), x)
- A154926 (2-exp(x), x)
- A216973 (x*exp(x), x)
- A093375 ((1+x)exp(x), x)
- A073107 (exp(x)/(1-x), x)
- A154921 (1/(2-exp(x)),x)
- A112227 (exp(-x^2),x)
- A122832 (exp(x(1+x)), x)
- A122833 (exp(-x(1+x)), x)
- A159834 (exp(-x-x^2/2), x)
- A110062 (exp(x(2+x)/2), x)
- A152150 (exp(x^2/2+x^4/4), x)
- A129652 (exp(x/(1-x)), x)
- A152151 ((1-x)exp(x/(1-x)), x)
- A116071 (exp(x*exp(x)), x)
- A215652 (exp(x*exp(-x)), x)
- A056857 (exp(exp(x)-1), x)
- A124323 (exp(exp(x)-1-x), x)
- A078937 (exp(2*exp(x)-2), x)
- A078938 (exp(3*exp(x)-3), x)
- A060081 (sech(x), tanh(x))
- A159830 (exp(exp(x)-1-2x), x)
- A152148 (exp(-x/2)(1-2x)^(-1/4), x)
- A067147 (exp(x^2), x)
- A099174 (exp(x^2/2), x)
- A119467 (cosh(x), x)
- A119879 (sech(x), x)
- A166335 (1+x*sinh(x), x)
- A166353 (1+x*tan(x/2), x)
- A166357 (1+x*atanh(x), x)
- A166378 (1+x^2*sec(x), x)
- A109187 (Bessel_I(0,2x), x)
- A097610 (Bessel_I(1,2x)/x, x)
- A091869 (exp(x)*Bessel_I(1,2x)/x, x)
- A091867 (exp(x)*(Bessel_I(0,2x)-Bessel_I(1,2x)), x)
- A124644 (exp(2x)*(Bessel_I(0,2x)-Bessel_I(1,2x)), x)
More exponential RA
- A049403 (1+x, x(1+x/2))
- A104556 (1-x, x(1-x/2))
- A176230 (1/sqrt(1-2x), x/(1-2x))
- A176231 (1/sqrt(1+2x), x/(1+2x))
- A001497 (1/sqrt(1-2x),1-sqrt(1-2x))
- A039683 (1/(1+2x), ln(1+2x)/2)
- A051141 (1/(1+3x), ln(1+3x)/3)
- A132393 (1/(1-x), ln(1/(1-x)))
- A154556 (exp(-x), x(1+x/2))
- A202995 (x*exp(x)/(exp(x)-1),x^2*exp(x)/(exp(x)-1))
- A059110 (exp(x/(1-x)), x/(1-x))
- A060821 (exp(-x^2), 2x)
- A049020 (exp(exp(x)-1), exp(x)-1)
- A111577 (exp(x), (exp(3x)-1)/3)
- A094816 (exp(x), ln(1/(1-x)))
- A176668 (exp(x), ln((exp(2x)+1)/2))
- A216294 (exp(x/(1-x)), ln(1/(1-x)))
- A028338 (1/sqrt(1-2x), ln(1/sqrt(1-2x)))
- A191916 ((2-2*cos(x))/x^2,(2-2*cos(x))/x)
- A199542 (x*cotan(x), x^2*cotan(x))
- A147308 (sech(x), asin(tanh(x)))
- A166318 (sech(2x), arctan(tanh(x)))
- A147309 (sec(x), ln(sec(x)+tan(x)))
- A166317 (sec(2x), atanh(tan(x)))
- A154602 (exp(sinh(x)*exp(x)), sinh(x)*exp(x))
- A161556 (1+(sqrt(Pi)/2)*x*exp(x^2/4)*ERF(x/2), x)
- A182826 (1/sqrt(1+2x+4x^2), arctan(sqrt(3)*x/(1+x))/sqrt(3))
- A088956,A182822-4,A185814,A185815,A185946,A185947,A185948,A191578,A215534