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

A047090 Sum{T(i,n-i): i=0,1,...,n}, array T as in A047089.

Original entry on oeis.org

1, 2, 3, 6, 12, 21, 42, 84, 155, 310, 620, 1176, 2352, 4704, 9071, 18142, 36284, 70717, 141434, 282868, 555268, 1110536, 2221072, 4381656, 8763312, 17526624, 34697879, 69395758, 138791516, 275467455, 550934910, 1101869820, 2191008499
Offset: 0

Views

Author

Clark Kimberling

Keywords

A047091 T(n,n), array T as in A047089.

Original entry on oeis.org

1, 1, 4, 11, 41, 139, 515, 1871, 7004, 26191, 99137, 376300, 1437105, 5505203, 21167781, 81602665, 315395985, 1221558683, 4740410605, 18426590090, 71736136773, 279654633032, 1091552835344, 4265355947434, 16684466516559
Offset: 0

Views

Author

Clark Kimberling

Keywords

A047092 T(n,n-1), array T as in A047089.

Original entry on oeis.org

1, 2, 4, 19, 60, 237, 850, 3252, 12168, 46497, 177052, 679745, 2612226, 10079515, 38961906, 150982379, 586067980, 2278999145, 8874984360, 34609160383, 135125659144, 528166347564, 2066529185116, 8093134894351
Offset: 1

Views

Author

Clark Kimberling

Keywords

A047093 T(n,n-2), array T as in A047089.

Original entry on oeis.org

1, 3, 8, 19, 98, 335, 1381, 5164, 20306, 77915, 303445, 1175121, 4574312, 17794125, 69379714, 270671995, 1057440462, 4134573755, 16182570293, 63389522371, 248511714532, 974976349772, 3827778946917, 15037653341387
Offset: 2

Views

Author

Clark Kimberling

Keywords

A047094 T(n,n-3), array T as in A047089.

Original entry on oeis.org

1, 4, 13, 38, 98, 531, 1912, 8138, 31418, 126393, 495376, 1962086, 7714610, 30417808, 119689616, 471372482, 1855574610, 7307585933, 28780361988, 113386055388, 446810002208, 1761249761801, 6944518447036, 27390257810022
Offset: 3

Views

Author

Clark Kimberling

Keywords

A047095 T(n,n+1), array T as in A047089.

Original entry on oeis.org

1, 2, 7, 22, 79, 278, 1021, 3752, 14023, 52640, 199248, 757360, 2892977, 11088266, 42640759, 164413606, 635490703, 2461411460, 9551605730, 37126976390, 144528973888, 563386487780, 2198826762318, 8591331622208
Offset: 0

Views

Author

Clark Kimberling

Keywords

A047096 T(n,n+2), array T as in A047089.

Original entry on oeis.org

1, 3, 11, 38, 139, 506, 1881, 7019, 26449, 100111, 381060, 1455872, 5583063, 21472978, 82810941, 320094718, 1239852777, 4811195125, 18700386300, 72792837115, 283731854748, 1107273926974, 4325975674774, 16918209611354
Offset: 0

Views

Author

Clark Kimberling

Keywords

A047097 T(n,n+3), array T as in A047089.

Original entry on oeis.org

1, 4, 16, 60, 228, 860, 3267, 12426, 47471, 181812, 698512, 2690086, 10384712, 40170182, 155681112, 604362074, 2349783665, 9148780570, 35665860725, 139202880860, 543887439194, 2127148912456, 8326877989146, 32623644335536
Offset: 0

Views

Author

Clark Kimberling

Keywords

A047098 a(n) = 2*binomial(3*n, n) - Sum_{k=0..n} binomial(3*n, k).

Original entry on oeis.org

1, 2, 8, 38, 196, 1062, 5948, 34120, 199316, 1181126, 7080928, 42860534, 261542752, 1607076200, 9934255472, 61732449648, 385393229460, 2415935640198, 15200964233864, 95962904716402, 607640599286276, 3858198001960438, 24559243585545644, 156692889782067712
Offset: 0

Views

Author

Clark Kimberling, Aug 15 1998

Keywords

Comments

T(2n,n), array T as in A047089. [Corrected Dec 08 2006]
Let B_3^+ denote the semigroup with presentation . Let D=aba be the 'fundamental word'. Then this sequence is also equal to the number of words in B_3^+ equal in B_3^+ to D^n, n >= 0. - Stephen P. Humphries, Jan 20 2004
In the language of Riordan arrays, row sums of (1/(1+x), x/(1+x)^3)^-1, where (1/(1+x), x/(1+x)^3) has general term (-1)^(n-k)*binomial(n+2k, 3k). - Paul Barry, May 09 2005
Hankel transform is 2^n*A051255(n) where A051255 is the Hankel transform of C(3n,n)/(2n+1). - Paul Barry, Jan 21 2007

Links

Crossrefs

Column k=2 of A213028.
Cf. A047089, A047099, A107026, A321957.

Programs

  • Maple
    A047098 := n -> 2*binomial(3*n, n)-add(binomial(3*n, k), k=0..n);
  • Mathematica
    Table[2Binomial[3n,n]-Sum[Binomial[3n,k],{k,0,n}],{n,0,35}] (* Harvey P. Dale, Jul 27 2011 *)
  • PARI
    a(n)=if(n<0,0,polcoeff((((1+10*x-2*x^2)+(1-4*x)*sqrt(1-4*x+x*O(x^n)))/2)^n,n))
  • PARI
    a(n)=if(n<0,0, 2*binomial(3*n,n)-sum(k=0,n,binomial(3*n,k)))

Formula

G.f. A(x)=y satisfies (8x-1)y^3-y^2+y+1=0. - Michael Somos, Jan 28 2004
Coefficient of x^n in ((1+10x-2x^2+(1-4x)^(3/2))/2)^n. - Michael Somos, Sep 25 2003
a(n) = Sum_{k = 0..n} A109971(k)*2^k; a(0) = 1, a(n) = Sum_{k = 0..n} 2^k*C(3n-k,n-k)*2*k/(3*n-k), n > 0. - Paul Barry, Jan 21 2007
Conjecture: 2*n*(2*n-1)*a(n) +(-71*n^2+112*n-48)*a(n-1) +3*(131*n^2-391*n+296)*a(n-2) -72*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Nov 30 2012
a(n) = A321957(n) + 2*binomial(3*n, n) - 8^n. - Peter Luschny, Nov 22 2018
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. - Peter Bala, Mar 05 2022

Extensions

Clark Kimberling, Dec 08 2006, changed "T(3n,2n)" to "T(2n,n)" in the comment line, but observes that some of the other comments seem to apply to the sequence T(3n,2n) rather than to the sequence T(2n,n).
Edited by N. J. A. Sloane, Dec 21 2006, replacing the old definition in terms of A047089 by an explicit formula supplied by Benoit Cloitre, Oct 25 2003.

A047120 Array T read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and touches the line y=x/4 only at lattice points.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 4, 2, 1, 1, 5, 10, 10, 6, 3, 1, 1, 6, 15, 20, 16, 6, 4, 1, 1, 7, 21, 35, 36, 22, 6, 5, 1, 1, 8, 28, 56, 71, 58, 28, 6, 6, 1, 1, 9, 36, 84, 127, 129, 86, 34, 12, 7, 1, 1, 10, 45, 120, 211, 256, 215, 120, 46, 19, 8, 1
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A047089 (with y=x/2), A047100 (with y=x/3).

Programs

  • Mathematica
    T[, 0] = 1; T[0, ] = 1; T[h_, k_] := T[h, k] = If[k-1 >= h/4 || k <= h/4, T[h, k-1], 0] + T[h-1, k];
Table[T[h - k, k], {h, 0, 11}, {k, h, 0, -1}]  // Flatten (* Jean-François Alcover, Mar 06 2019 *)
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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