List of Riordan arrays

(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

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

Multidimensional tables

A107027 (1/(1+x), x/(1+x)^k), k>=0
A107030 (1/(1+x), x/(1+x)^k), k>=0

List of Riordan arrays

Contents

How to expand Riordan arrays

Ordinary RA

Lagrange subgroup R(1,h(x))

Appell subgroup R(d(x),x)

Bell subgroup R(d(x),xd(x)) (convolution triangles)

Hitting time arrays R(xh'(x)/h(x),xh(x))

More Ordinary RA (proper or improper)

h(x) not a fraction

h(x) denominator a power of 1-x

h(x) denominator a power of 1+x

More rational h(x)

Inverses of rational R

In terms of sqrt() only

u = sqrt(1-4x)

u = sqrt(1-2x-3^2)

u = sqrt(1-6x+x^2)

Mixed or other poynomials in sqrt()

Using c(x), M(x),...

Inverses without known d(x),h(x)

Exponential Riordan arrays

Lagrange subgroup

Appell subgroup

More exponential RA

Multidimensional tables

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