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 1-5 of 5 results.

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 9, 12, 6, 5, 16, 36, 48, 24, 6, 25, 80, 180, 240, 120, 7, 36, 150, 480, 1080, 1440, 720, 8, 49, 252, 1050, 3360, 7560, 10080, 5040, 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320, 10, 81, 576, 3528, 18144, 75600, 241920, 544320
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 30 2014

Keywords

Comments

row(0) = {1}; row(n+1) = row(n) multiplied by n and prepended with (n+1);
A111063(n+1) = sum of n-th row;
T(2*n,n) = A002690(n), central terms;
T(n,0) = n + 1;
T(n,1) = A000290(n), n > 0;
T(n,2) = A011379(n-1), n > 1;
T(n,3) = A047927(n), n > 2;
T(n,4) = A192849(n-1), n > 3;
T(n,5) = A000142(5) * A027810(n-5), n > 4;
T(n,6) = A000142(6) * A027818(n-6), n > 5;
T(n,7) = A000142(7) * A056001(n-7), n > 6;
T(n,8) = A000142(8) * A056003(n-8), n > 7;
T(n,9) = A000142(9) * A056114(n-9), n > 8;
T(n,n-10) = 11 * A051431(n-10), n > 9;
T(n,n-9) = 10 * A049398(n-9), n > 8;
T(n,n-8) = 9 * A049389(n-8), n > 7;
T(n,n-7) = 8 * A049388(n-7), n > 6;
T(n,n-6) = 7 * A001730(n), n > 5;
T(n,n-5) = 6 * A001725(n), n > 5;
T(n,n-4) = 5 * A001720(n), n > 4;
T(n,n-3) = 4 * A001715(n), n > 2;
T(n,n-2) = A070960(n), n > 1;
T(n,n-1) = A052849(n), n > 0;
T(n,n) = A000142(n);
T(n,k) = A137948(n,k) * A007318(n,k), 0 <= k <= n.

Examples

			.  0:   1;
.  1:   2,  1;
.  2:   3,  4,   2;
.  3:   4,  9,  12,    6;
.  4:   5, 16,  36,   48,    24;
.  5:   6, 25,  80,  180,   240,   120;
.  6:   7, 36, 150,  480,  1080,  1440,    720;
.  7:   8, 49, 252, 1050,  3360,  7560,  10080,   5040;
.  8:   9, 64, 392, 2016,  8400, 26880,  60480,  80640,  40320;
.  9:  10, 81, 576, 3528, 18144, 75600, 241920, 544320, 725760, 362880.

Links

Crossrefs

Cf. A111063 (row sums), A240993 (row products), A002690 (central terms).
Cf. A000290, A011379, A027810, A027818, A047927, A056001, A056003, A056114, A192849.
Cf. A000142, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431, A052849, A070960.
Cf. A007318, A137948.

Programs

  • Haskell
    a245334 n k = a245334_tabl !! n !! k
a245334_row n = a245334_tabl !! n
a245334_tabl = iterate (\row@(h:_) -> (h + 1) : map (* h) row) [1]
  • Mathematica
    Table[(n!)/((n - k)!)*(n + 1 - k), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 10 2017 *)

Formula

T(n,k) = n!*(n+1-k)/(n-k)!. - Werner Schulte, Sep 09 2017

A093644 (9,1) Pascal triangle.

Original entry on oeis.org

1, 9, 1, 9, 10, 1, 9, 19, 11, 1, 9, 28, 30, 12, 1, 9, 37, 58, 42, 13, 1, 9, 46, 95, 100, 55, 14, 1, 9, 55, 141, 195, 155, 69, 15, 1, 9, 64, 196, 336, 350, 224, 84, 16, 1, 9, 73, 260, 532, 686, 574, 308, 100, 17, 1, 9, 82, 333, 792, 1218, 1260, 882, 408, 117, 18, 1, 9, 91, 415
Offset: 0

Views

Author

Wolfdieter Lang, Apr 22 2004

Keywords

Comments

The array F(9;n,m) gives in the columns m>=1 the figurate numbers based on A017173, including the 11-gonal numbers A051682 (see the W. Lang link).
This is the ninth member, d=9, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-5, for d=1..8.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m is G(z,x) = (1+8*z)/(1-(1+x)*z).
The SW-NE diagonals give A022099(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 8. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
Triangle T(n,k), read by rows, given by (9,-8,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 10 2011

Examples

			Triangle begins
  [1];
  [9,  1];
  [9, 10,  1];
  [9, 19, 11,  1];
  ...

References

  • Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
  • Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Links

Crossrefs

Row sums: A020714(n-1), n >= 1, 1 for n=0, alternating row sums are 1 for n=0, 8 for n=2 and 0 otherwise.
The column sequences give for m=1..9: A017173, A051682 (11-gonal), A007586, A051798, A051879, A050405, A052206, A056117, A056003.
Cf. A093645 (d=10).

Programs

  • Haskell
    a093644 n k = a093644_tabl !! n !! k
a093644_row n = a093644_tabl !! n
a093644_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [9, 1]
-- Reinhard Zumkeller, Aug 31 2014
  • Mathematica
    Join[{1},Table[Binomial[n,k]+8Binomial[n-1,k],{n,20},{k,0,n}]//Flatten] (* Harvey P. Dale, Aug 17 2024 *)

Formula

a(n, m) = F(9;n-m, m) for 0 <= m <= n, otherwise 0, with F(9;0, 0)=1, F(9;n, 0)=9 if n >= 1 and F(9;n, m):=(9*n+m)*binomial(n+m-1, m-1)/m if m >= 1.
Recursion: a(n, m)=0 if m > n, a(0, 0)= 1; a(n, 0)=9 if n >= 1; a(n, m) = a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (1+8*x)/(1-x)^(m+1), m >= 0.
T(n, k) = C(n, k) + 8*C(n-1, k). - Philippe Deléham, Aug 28 2005
Row n: Expansion of (9+x)*(1+x)^(n-1), n > 0. - Philippe Deléham, Oct 10 2011
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(9 + 19*x + 11*x^2/2! + x^3/3!) = 9 + 28*x + 58*x^2/2! + 100*x^3/3! + 155*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014
G.f.: (-1-8*x)/(-1+x+x*y). - R. J. Mathar, Aug 11 2015

A254142 a(n) = (9*n+10)*binomial(n+9,9)/10.

Original entry on oeis.org

1, 19, 154, 814, 3289, 11011, 32032, 83512, 199342, 442442, 923780, 1830764, 3468374, 6317234, 11113784, 18958808, 31461815, 50930165, 80613390, 125014890, 190285095, 284712285, 419329560, 608658960, 871616460, 1232604516, 1722822024, 2381824984
Offset: 0

Views

Author

Bruno Berselli, Jan 26 2015

Keywords

Comments

Partial sums of A056003.
If n is of the form 8*k+2*(-1)^k-1 or 8*k+2*(-1)^k-2 then a(n) is odd.

Links

Crossrefs

Cf. A000581, A001287, A056003.
Cf. sequences of the type (k*n+k+1)*binomial(n+k,k)/(k+1): A000217 (k=1), A000330 (k=2), A001296 (k=3), A034263 (k=4), A051946 (k=5), A034265 (k=6), A034266 (k=7), A056122 (k=8), this sequence (k=9).

Programs

  • GAP
    List([0..30], n-> (9*n+10)*Binomial(n+9,9)/10); # G. C. Greubel, Aug 28 2019
  • Magma
    [(9*n+10)*Binomial(n+9,9)/10: n in [0..30]];
  • Maple
    seq((9*n+10)*binomial(n+9,9)/10, n=0..30); # G. C. Greubel, Aug 28 2019
  • Mathematica
    Table[(9n+10)Binomial[n+9, 9]/10, {n, 0, 30}]
  • PARI
    vector(30, n, n--; (9*n+10)*binomial(n+9, 9)/10)
  • Sage
    [(9*n+10)*binomial(n+9,9)/10 for n in (0..30)]

Formula

G.f.: (1 + 8*x)/(1-x)^11.
a(n) = Sum_{i=0..n} (i+1)*A000581(i+8).
a(n+1) = 8*A001287(n+10) + A001287(n+11).

A056114 Expansion of (1+9*x)/(1-x)^11.

Original entry on oeis.org

1, 20, 165, 880, 3575, 12012, 35035, 91520, 218790, 486200, 1016158, 2015520, 3821090, 6963880, 12257850, 20920064, 34730575, 56241900, 89049675, 138138000, 210315105, 314757300, 463681725, 673171200, 964177500, 1363732656, 1906401420, 2636011840, 3607704980
Offset: 0

Views

Author

Barry E. Williams, Jun 12 2000

Keywords

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Cf. A000142, A007318, A056003, A245334.

Programs

  • GAP
    List([0..40], n-> (n+1)*Binomial(n+9, 9)); # G. C. Greubel, Jan 18 2020
  • Haskell
    a056114 n = (n + 1) * a007318' (n + 9) 9
-- Reinhard Zumkeller, Aug 31 2014
  • Magma
    [(n+1)*Binomial(n+9, 9): n in [0..40]]; // G. C. Greubel, Jan 18 2020
  • Maple
    a:=n->(sum((numbcomp(n,10)), j=10..n)):seq(a(n), n=10..34); # Zerinvary Lajos, Aug 26 2008
  • Mathematica
    CoefficientList[Series[(1+9x)/(1-x)^11,{x,0,40}],x] (* or *) LinearRecurrence[ {11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,20,165,880,3575,12012,35035, 91520,218790,486200,1016158},40] (* Harvey P. Dale, Jun 05 2018 *)
  • PARI
    vector(41, n, n*binomial(n+8, 9) ) \\ G. C. Greubel, Jan 18 2020
  • Sage
    [(n+1)*binomial(n+9, 9) for n in (0..40)] # G. C. Greubel, Jan 18 2020

Formula

a(n) = (n+1)*binomial(n+9, 9).
G.f.: (1+9*x)/(1-x)^11.
a(n) = A245334(n+9,9)/A000142(9). - Reinhard Zumkeller, Aug 31 2014
From G. C. Greubel, Jan 18 2020: (Start)
a(n) = 10*binomial(n+10,10) - 9*binomial(n+9,9).
E.g.f.: (9! +6894720*x +22861440*x^2 +26853120*x^3 +14605920*x^4 + 4191264*x^5 +677376*x^6 +63072*x^7 +3321*x^8 +91*x^9 +x^10)*exp(x)/9!. (End)
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 3*Pi^2/2 - 1077749/78400.
Sum_{n>=0} (-1)^n/a(n) = 3*Pi^2/4 - 24576*log(2)/35 + 37652469/78400. (End)

A056128 a(n) = (9*n + 11)*binomial(n+10, 10)/11.

Original entry on oeis.org

1, 20, 174, 988, 4277, 15288, 47320, 130832, 330174, 772616, 1696396, 3527160, 6995534, 13312768, 24426552, 43385360, 74847175, 125777340, 206390730, 331405620, 521690715, 806403000, 1225732560, 1834391520, 2706007980, 3938612496, 5661434520, 8043259504
Offset: 0

Views

Author

Barry E. Williams, Jul 08 2000

Keywords

Links

Crossrefs

Cf. A056003.

Programs

  • GAP
    List([0..30], n-> (9*n+11)*Binomial(n+10,10)/11 ); # G. C. Greubel, Jan 18 2020
  • Magma
    [((9*n+11)*Binomial(n+10,10))/11: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014
  • Maple
    seq( (9*n+11)*binomial(n+10, 10)/11, n=0..30); # G. C. Greubel, Jan 18 2020
  • Mathematica
    CoefficientList[Series[(1+8x)/(1-x)^12, {x,0,40}], x] (* Vincenzo Librandi, Jul 30 2014 *)
LinearRecurrence[{12,-66,220,-495,792,-924,792,-495,220,-66,12,-1}, {1,20,174, 988,4277,15288,47320,130832,330174,772616,1696396,3527160}, 40] (* Harvey P. Dale, Jan 14 2015 *)
  • PARI
    vector(31, n, (9*n-2)*binomial(n+9,10)/11 ) \\ G. C. Greubel, Jan 18 2020
  • Sage
    [(9*n+11)*binomial(n+10,10)/11 for n in (0..30)] # G. C. Greubel, Jan 18 2020

Formula

a(n) = (9*n + 11)*binomial(n+10, 10)/11.
G.f.: (1+8*x)/(1-x)^12.
a(n) = 9*binomial(n+11,11) - 8*binomial(n+10,10). - G. C. Greubel, Jan 18 2020

Extensions

New name, from existing formula, added by G. C. Greubel, Jan 18 2020
Showing 1-5 of 5 results.

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