cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A112708 Row sums of triangle A112707 (partial sums of Catalan numbers multiplied by powers of negative numbers).

Original entry on oeis.org

1, 2, 2, 3, 4, -7, 68, -285, 553, 8052, -142982, 1505079, -11345907, 38452956, 672559916, -18995038455, 301298528898, -3473588012773, 23630888402370, 167786255489085, -10244305495490866, 242112502660674867, -4105880426369854764, 48839647963108358039
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Crossrefs

Cf. A112707, A112709 (unsigned row sums).

Formula

a(n) = Sum_{m=0..n} A112707(n, m), n>=0.

A112709 Unsigned row sums of triangle A112707 (partial sums of Catalan numbers multiplied by powers of negative numbers).

Original entry on oeis.org

1, 2, 2, 5, 14, 65, 376, 2613, 20761, 184756, 1816330, 19526217, 227588213, 2855033988, 38311846704, 547128228043, 8279365967056, 132259968568859, 2223048616707842, 39199883209336095, 723269314782624612
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Crossrefs

Cf. A112707, A112708 (signed row sums).

Formula

a(n) = Sum_{m=0..n} |A112707(n, m)|, for n >= 0.

A062992 Row sums of unsigned triangle A062991.

Original entry on oeis.org

1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173
Offset: 0

Views

Author

Wolfdieter Lang, Jul 12 2001

Keywords

Comments

a(n) = N(2; n,x=-1), with the polynomials N(2; n,x) defined in A062991.

Links

Crossrefs

Cf. A112707 (c(n, -m) triangle). Here m=2 is used. Row sums of A234950.
Cf. A000108, A062991, A064062, A119259.

Programs

  • Haskell
    a062992 = sum . a234950_row  -- Reinhard Zumkeller, Jan 12 2014
  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1-2*x-Sqrt(1-8*x))/(2*x+2*x^2) )); // G. C. Greubel, Sep 27 2024
  • Mathematica
    Table[2*Sum[(-1)^j*Binomial[2*n-2*j,n-j]/(n-j+1)*2^(n-j), {j,0,n}]-(-1)^n,{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
  • PARI
    a(n)=polcoeff((1-2*x-sqrt(1-8*x+x^2*O(x^n)))/(2*x+2*x^2),n)
  • PARI
    a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x)^2+O(x^(n+2))),n+1)) \\ Ralf Stephan
  • Sage
    def a(n): return hypergeometric([-n, n+1], [-n-1], 2)
[a(n).hypergeometric_simplify() for n in range(21)] # Peter Luschny, Nov 30 2014

Formula

a(n) = (-1)^(n+1) + 2*Sum_{j = 0..n} (-1)^j*C(n-j)*2^(n-j) with C(n) := A000108(n) (Catalan).
G.f.: A(x) = (2*c(2*x) - 1)/(1 + x) with c(x) the g.f. of A000108.
a(n) = (1/(n+1)) * Sum_{k = 0..n} binomial(2*n+2, n-k)*binomial(n+k, k). - Paul Barry, May 11 2005
Rewritten: a(n) = (1 - 2*c(n, -2))*(-1)^(n+1), n >= , with c(n, x) := Sum_{k = 0..n} C(k)*x^k and C(k) := A000108(k) (Catalan). - Wolfdieter Lang, Oct 31 2005
Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ 2^(3*n+4)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012
a(n) = hypergeometric([-n, n+1], [-n-1], 2). - Peter Luschny, Nov 30 2014
G.f.: A(x) = exp( Sum_{n >= 1} A119259(n)*x^n/n ). - Peter Bala, Jun 08 2023

A112710 Partial sums of Catalan numbers A000108 multiplied by powers of -3.

Original entry on oeis.org

1, -2, 16, -119, 1015, -9191, 87037, -851186, 8531044, -87167702, 904619302, -9509144240, 101036961052, -1083385595648, 11708378016712, -127402051267703, 1394629704432367, -15347719608268403, 169699299111055897, -1884322608673443833, 21003350078068124587
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Comments

Fourth column (m=3) of triangle A112707 (partial sums of Catalan numbers multiplied by powers of nonpositive numbers).

Crossrefs

Cf. A064306 (unsigned m=2 case).

Formula

a(n)= A112707(n+3, 3), n>=0.
G.f.: c(-3*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
G.f. for unsigned sequence: c(3*x)/(1+x), hence |a(n)| is the convolution of the sequences {C(n)*3^n} and {(-1)^n}, where C(n):=A000108(n).

A112711 Partial sums of Catalan numbers A000108 multiplied by powers of -4.

Original entry on oeis.org

1, -3, 29, -291, 3293, -39715, 500957, -6527779, 87188701, -1187355427, 16424527069, -230141827875, 3259720426717, -46595454638883, 671319066305757, -9738441487391523, 142121594825368797, -2085158937761782563, 30737922595101500637, -455043684091275090723
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Comments

Fifth column (m=4) of triangle A112707 (partial sums of Catalan numbers multiplied by powers of nonpositive numbers).

Crossrefs

Cf. A112710 (m=3).

Formula

a(n)= A112707(n+4, 4), n>=0.
G.f.: c(-4*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
G.f. for unsigned sequence: c(4*x)/(1+x), hence |a(n)| is the convolution of the sequences {C(n)*4^n} and {(-1)^n}, where C(n):=A000108(n).

A113264 Partial sums of Catalan numbers A000108 multiplied by powers of -5.

Original entry on oeis.org

1, -4, 46, -579, 8171, -123079, 1939421, -31576204, 527017546, -8969076204, 155054361296, -2715355794954, 48068823892546, -858791527669954, 15464694800455046, -280398494896810579, 5114753787818033171, -93796371395287435579, 1728250671451392251921
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Comments

Sixth column (m=5) of triangle A112707 (partial sums of Catalan numbers multiplied by powers of nonpositive numbers).

Crossrefs

Cf. A112711 (m=4).

Formula

a(n)= A112707(n+5, 5), n>=0.
G.f.: c(-5*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
G.f. for unsigned sequence: c(5*x)/(1+x), hence |a(n)| is the convolution of the sequences {C(n)*5^n} and {(-1)^n}, where C(n):=A000108(n).

A113265 Partial sums of Catalan numbers A000108 multiplied by powers of -6.

Original entry on oeis.org

1, -5, 67, -1013, 17131, -309461, 5849131, -114243413, 2287607467, -46710150485, 968879741611, -20358507992405, 432438339283627, -9270351245202773, 200309903779703467, -4358060643012007253, 95389812498547780267, -2099063396615767545173
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Comments

Seventh column (m=6) of triangle A112707 (partial sums of Catalan numbers multiplied by powers of nonpositive numbers).

Crossrefs

Cf. A113264 (m=5).

Formula

a(n)= A112707(n+6, 6), n>=0.
G.f.: c(-6*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
G.f. for unsigned sequence: c(6*x)/(1+x), hence |a(n)| is the convolution of the sequences {C(n)*6^n} and {(-1)^n}, where C(n):=A000108(n).

A113266 Partial sums of Catalan numbers A000108 multiplied by powers of -7.

Original entry on oeis.org

1, -6, 92, -1623, 31991, -673903, 14855765, -338444182, 7905221248, -188294015986, 4556160266218, -111682969647780, 2767470863606632, -69211374967753668, 1744655539982525892, -44282217426880817943, 1130756774785983371727
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Comments

Eighth column (m=7) of triangle A112707 (partial sums of Catalan numbers multiplied by powers of nonpositive numbers).

Crossrefs

Cf. A113265 (m=6).

Formula

a(n)= A112707(n+7, 7), n>=0.
G.f.: c(-7*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
G.f. for unsigned sequence: c(7*x)/(1+x), hence |a(n)| is the convolution of the sequences {C(n)*7^n} and {(-1)^n}, where C(n):=A000108(n).

A113267 Partial sums of Catalan numbers A000108 multiplied by powers of -8.

Original entry on oeis.org

1, -7, 121, -2439, 54905, -1321351, 33281657, -866396551, 23125022329, -629441571207, 17405126104697, -487562768820615, 13806913025988217, -394606681111406983, 11367704830045574777, -329739328993506896263, 9622560010799553435257
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Comments

Ninth column (m=8) of triangle A112707 (partial sums of Catalan numbers multiplied by powers of nonpositive numbers).

Crossrefs

Cf. A113266 (m=7).

Formula

a(n)= A112707(n+8, 8), n>=0.
G.f.: c(-8*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
G.f. for unsigned sequence: c(8*x)/(1+x), hence |a(n)| is the convolution of the sequences {C(n)*8^n} and {(-1)^n}, where C(n):=A000108(n).

A113268 Partial sums of Catalan numbers A000108 multiplied by powers of -9.

Original entry on oeis.org

1, -8, 154, -3491, 88363, -2391695, 67758517, -1984135184, 59572675846, -1824065741672, 56739965057524, -1788027005117150, 56960705737368622, -1831391418128245478, 59351217395117651362, -1936731395137029733043, 63581744945624043237427
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Comments

Tenth column (m=9) of triangle A112707 (partial sums of Catalan numbers multiplied by powers of nonpositive numbers).

Crossrefs

Cf. A113267 (m=8).

Formula

a(n)= A112707(n+9, 9), n>=0.
G.f.: c(-9*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
G.f. for unsigned sequence: c(9*x)/(1+x), hence |a(n)| is the convolution of the sequences {C(n)*9^n} and {(-1)^n}, where C(n):=A000108(n).
Showing 1-10 of 11 results. Next

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