A051798 -id:A051798 - cp's OEIS frontend

A051879 Partial sums of A051798.

1, 14, 69, 224, 574, 1260, 2478, 4488, 7623, 12298, 19019, 28392, 41132, 58072, 80172, 108528, 144381, 189126, 244321, 311696, 393162, 490820, 606970, 744120, 904995, 1092546, 1309959, 1560664

Offset: 0

Barry E. Williams, Dec 14 1999

Convolution of triangular numbers (A000217) and 11-gonal numbers (A051682). [Bruno Berselli, Jul 21 2015]

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-16.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A051798; A000217, A051682.
Cf. A093644((9, 1) Pascal, column m=5).
Cf. A050405.

Accumulate[Table[(n+1)(n+2)(n+3)(9n+4)/24,{n,0,40}]] (* Harvey P. Dale, Aug 19 2012 *)

a(n) = C(n+4, 4)*(9n+5)/5.

G.f.: (1+8*x)/(1-x)^6.

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

Wolfdieter Lang, Apr 22 2004

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

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

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.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Wolfdieter Lang, First 10 rows and array of figurate numbers .

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).

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

Join[{1},Table[Binomial[n,k]+8Binomial[n-1,k],{n,20},{k,0,n}]//Flatten] (* Harvey P. Dale, Aug 17 2024 *)

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

A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

Original entry on oeis.org

0, 1, 16, 70, 200, 455, 896, 1596, 2640, 4125, 6160, 8866, 12376, 16835, 22400, 29240, 37536, 47481, 59280, 73150, 89320, 108031, 129536, 154100, 182000, 213525, 248976, 288666, 332920, 382075, 436480, 496496, 562496, 634865, 714000, 800310, 894216, 996151

Offset: 0

Bruno Berselli, Dec 08 2012

Partial sums of A172073.

Apart from 0, all terms are in A135021: a(n) = A135021(A034856(n+1)) with n>0.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index to sequences related to pyramidal numbers.

Cf. A135021, A172073.
Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):
k= 2 (A000027 ): A000292;
k= 3 (A000217 ): A000332 (after the third term);
k= 4 (A000290 ): A002415 (after the first term);
k= 5 (A000326 ): A001296;
k= 6 (A000384*): A002417;
k= 7 (A000566 ): A002418;
k= 8 (A000567*): A002419;
k= 9 (A001106*): A051740;
k=10 (A001107*): A051797;
k=11 (A051682*): A051798;
k=12 (A051624*): A051799;
k=13 (A051865*): A055268.
Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

A051866:=func; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];

I:=[0,1,16,70,200]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]
CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x*(1+11*x)/(1-x)^5.
a(n) = n*(n+1)*(n+2)*(3*n-2)/6.
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)

A088941 a(n)=12sum(1<=i<=j<=k<=n,ij/k).

Original entry on oeis.org

0, 12, 54, 154, 349, 685, 1217, 2009, 3134, 4674, 6720, 9372, 12739, 16939, 22099, 28355, 35852, 44744, 55194, 67374, 81465, 97657, 116149, 137149, 160874, 187550, 217412, 250704, 287679, 328599, 373735, 423367, 477784, 537284, 602174, 672770

Offset: 0

Benoit Cloitre, Oct 25 2003

Always an integer.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A088942, A088943.

A051798(n) - 1.

LinearRecurrence[{5,-10,10,-5,1},{0,12,54,154,349},40] (* Harvey P. Dale, May 01 2014 *)

a(n)=if(n<0,0,a(n)=n/24*(9*n^3 + 58*n^2 + 123*n + 98))

a(n)=(n/24)*(9*n^3 + 58*n^2 + 123*n + 98).

G.f.: x*(x^3-4*x^2+6*x-12) / (x-1)^5. - Colin Barker, Jun 15 2013

A104728 Triangle T(n,k) = (k-1-n)(k-2-n)(k-2+2*n)/2 read by rows, 1<=k<=n.

Original entry on oeis.org

1, 9, 4, 30, 18, 7, 70, 48, 27, 10, 135, 100, 66, 36, 13, 231, 180, 130, 84, 45, 16, 364, 294, 225, 160, 102, 54, 19, 540, 448, 357, 270, 190, 120, 63, 22, 765, 648, 532, 420, 315, 220, 138, 72, 25, 1045, 900, 756, 616, 483, 360, 250, 156, 81, 28, 1386, 1210, 1035, 864, 700, 546, 405, 280, 174, 90, 31

Offset: 1

Gary W. Adamson, Mar 20 2005

The triangle is defined as the matrix product A * B, A = [1; 1, 4; 1, 4, 7;...]; B = [1; 2, 1; 3, 2, 1;...]; both infinite lower triangular matrices with the rest of the terms zeros.

			The first few rows of the triangle are:
1;
9,    4;
30,   18,   7;
70,   48,   27,   10;
135,  100,  66,   36,   13;
231,  180,  130,  84,   45,  16;
364,  294,  225,  160,  102, 54,  19;
540,  448,  357,  270,  190, 120, 63,  22;
765,  648,  532,  420,  315, 220, 138, 72,  25;
1045, 900,  756,  616,  483, 360, 250, 156, 81,  28;
1386, 1210, 1035, 864,  700, 546, 405, 280, 174, 90,  31;
1794, 1584, 1375, 1170, 972, 784, 609, 450, 310, 192, 99, 34, etc.

Cf. A051798 (row sums), A007586, A002414 (column 1).

A104728 := proc(n)
        (k-1-n)*(k-2-n)*(k-2+2*n)/2 ;
end proc:
seq(seq(A104728(n,k),k=1..n),n=1..14) ; # R. J. Mathar, Nov 07 2011

Table[(k-1-n)(k-2-n)(k-2+2n)/2,{n,20},{k,n}]//Flatten (* Harvey P. Dale, Dec 25 2018 *)

Name contributed by R. J. Mathar, Nov 07 2011

A349147 Triangle T(n,m) read by rows: the sum of runs of all sequences arranging n objects of one type and m objects of another type.

Original entry on oeis.org

1, 1, 4, 1, 7, 18, 1, 10, 34, 80, 1, 13, 55, 155, 350, 1, 16, 81, 266, 686, 1512, 1, 19, 112, 420, 1218, 2982, 6468, 1, 22, 148, 624, 2010, 5412, 12804, 27456, 1, 25, 189, 885, 3135, 9207, 23595, 54483, 115830, 1, 28, 235, 1210, 4675, 14872, 41041, 101530, 230230, 486200, 1, 31

Offset: 0

R. J. Mathar, Nov 08 2021

			The triangle starts
  1,
  1,  4,
  1,  7,  18,
  1, 10,  34,   80,
  1, 13,  55,  155,  350,
  1, 16,  81,  266,  686,  1512,
  1, 19, 112,  420, 1218,  2982,  6468,
  1, 22, 148,  624, 2010,  5412, 12804,  27456,
  1, 25, 189,  885, 3135,  9207, 23595,  54483, 115830,
  1, 28, 235, 1210, 4675, 14872, 41041, 101530, 230230, 486200,
  1, 31, 286, 1606, 6721, 23023, 68068, 179608, 432718, 967538, 2032316
For n=m=1 the sequences are ab (2 runs) and ba (2 runs), so T(1,1)=2+2=4.
For n=1, m=2 the sequences are aab (2 runs), aba (3 runs), baa (2 runs), so T(1,2)=2+3+2=7.
For n=m=2 the sequences are aabb (2 runs), abab (4 runs), abba (3 runs), baab (3 runs), baba (4 runs), bbaa (2 runs), so T(2,2) = 2+4+3+3+4+2=18.

R. J. Mathar, The Number of Runs of Words on a 2-letter Alphabet (2021)

Cf. A016777 (row/col 1), A000566 (row/col 2), A007584 (row/col 3), A051798 (row/col 4).

Diagonal gives A037965(n+1).

T(n,m) = T(m,n).
Sum_{m=0..n} T(n,m) = A000917(n-1) + A000984(n) = 1, 5, 26, 125, 574, ... - R. J. Mathar, Nov 09 2021
T(n,m) = binomial(n+m,n)*(2*n*m+n+m)/(n+m) for n+m >= 1.

cp's OEIS Frontend

A051879 Partial sums of A051798.

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A093644 (9,1) Pascal triangle.

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A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

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Magma

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A088941 a(n)=12*sum(1<=i<=j<=k<=n,i*j/k).

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A104728 Triangle T(n,k) = (k-1-n)*(k-2-n)*(k-2+2*n)/2 read by rows, 1<=k<=n.

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Maple

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A349147 Triangle T(n,m) read by rows: the sum of runs of all sequences arranging n objects of one type and m objects of another type.

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Formula

A088941 a(n)=12sum(1<=i<=j<=k<=n,ij/k).

A104728 Triangle T(n,k) = (k-1-n)(k-2-n)(k-2+2*n)/2 read by rows, 1<=k<=n.