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List of Riordan arrays

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(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

Ordinary RA

Lagrange subgroup R(1,h(x))

Appell subgroup R(d(x),x)

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))

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)

  • 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)

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),...

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

Exponential Riordan arrays

Lagrange subgroup

Appell subgroup

More exponential RA

Multidimensional tables

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