A005585 -id:A005585 - cp's OEIS frontend

A054334 1/512 of 11th unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).

1, 12, 77, 352, 1287, 4004, 11011, 27456, 63206, 136136, 277134, 537472, 999362, 1790712, 3105322, 5230016, 8580495, 13748020, 21559395, 33153120, 50075025, 74397180, 108864405, 157073280, 223689180, 314707536, 437766252, 602516992, 821063892, 1108479152

Offset: 0

Wolfdieter Lang

Partial sums of A054333.
If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-11) is the number of 11-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
10-dimensional square numbers, ninth partial sums of binomial transform of [1,2,0,0,0,...]. a(n)=sum{i=0,n,C(n+9,i+9)*b(i)}, where b(i)=[1,2,0,0,0,...]. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
2*a(n) is number of ways to place 9 queens on an (n+9) X (n+9) chessboard so that they diagonally attack each other exactly 36 times. The maximal possible attack number, p=binomial(k,2)=36 for k=9 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph. - Antal Pinter, Dec 27 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Milan Janjic, Two Enumerative Functions.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A053120, A054333.

Cf. A005585, A040977, A050486, A053347, A054333.

List([0..30], n -> (2*n+10)*Binomial(n+9, 9)/10); # G. C. Greubel, Dec 02 2018

[(2*n+10)*Binomial(n+9, 9)/10: n in [0..40]]; // G. C. Greubel, Dec 02 2018

Table[(2*n + 10)*Binomial[n + 9, 9]/10, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,12,77,352,1287,4004,11011,27456,63206,136136,277134},30] (* Harvey P. Dale, May 11 2025 *)

vector(40, n, n--; (2*n+10)*binomial(n+9, 9)/10) \\ G. C. Greubel, Dec 02 2018

[(2*n+10)*binomial(n+9, 9)/10 for n in range(40)] # G. C. Greubel, Dec 02 2018

a(n) = (2*n+10)*binomial(n+9, 9)/10 = ((-1)^n)*A053120(2*n+10, 10)/2^9.
G.f.: (1+x)/(1-x)^11.
a(n) = 2*binomial(n+10, 10) - binomial(n+9, 9). - Paul Barry, Mar 04 2003
a(n) = binomial(n+9,9) + 2*binomial(n+9,10). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
a(n) = binomial(n+9,9)*(n+5)/5. - Antal Pinter, Dec 27 2015
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=0} 1/a(n) = 525*Pi^2 - 1160419/224.
Sum_{n>=0} (-1)^n/a(n) = 525*Pi^2/2 - 82875/32. (End)

A054333 1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).

Original entry on oeis.org

1, 11, 65, 275, 935, 2717, 7007, 16445, 35750, 72930, 140998, 260338, 461890, 791350, 1314610, 2124694, 3350479, 5167525, 7811375, 11593725, 16921905, 24322155, 34467225, 48208875, 66615900, 91018356, 123058716, 164750740

Offset: 0

Barry E. Williams, Wolfdieter Lang, Mar 15 2000

If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-10) is the number of 10-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
9-dimensional square numbers, eighth partial sums of binomial transform of [1,2,0,0,0,...]. a(n)=sum{i=0,n,C(n+8,i+8)*b(i)}, where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
2*a(n) is number of ways to place 8 queens on an (n+8) X (n+8) chessboard so that they diagonally attack each other exactly 28 times. The maximal possible attack number, p=binomial(k,2) =28 for k=8 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph. - Antal Pinter, Dec 27 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Milan Janjic, Two Enumerative Functions.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
Index entries for sequences related to Chebyshev polynomials.

Partial sums of A053347. Cf. A053120, A000581.
Cf. A005585, A040977, A050486, A053347. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009
Cf. A111125, fifth column (s=4, without leading zeros). - Wolfdieter Lang, Oct 18 2012

List([0..30],n->(2*n+9)*Binomial(n+8,8)/9); # Muniru A Asiru, Dec 06 2018

[Binomial(n+8,8)+2*Binomial(n+8,9): n in [0..40]]; // Vincenzo Librandi, Feb 14 2016

LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {1, 11, 65, 275, 935, 2717, 7007, 16445, 35750, 72930}, 30] (* Vincenzo Librandi, Feb 14 2016 *)

vector(40, n, n--; (2*n+9)*binomial(n+8, 8)/9) \\ G. C. Greubel, Dec 02 2018

[(2*n+9)*binomial(n+8, 8)/9 for n in range(40)] # G. C. Greubel, Dec 02 2018

a(n) = (2*n+9)*binomial(n+8, 8)/9 = ((-1)^n)*A053120(2*n+9, 9)/2^8.
G.f.: (1+x)/(1-x)^10.
a(n) = 2*C(n+9, 9) - C(n+8, 8). - Paul Barry, Mar 04 2003
a(n) = C(n+8,8) + 2*C(n+8,9). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
E.g.f.: (1/362880)*exp(x)*(362880 + 3628800*x + 7983360*x^2 + 6773760*x^3 + 2751840*x^4 + 592704*x^5 + 70560*x^6 + 4608*x^7 + 153*x^8 + 2*x^9). - Stefano Spezia, Dec 03 2018
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=0} 1/a(n) = 294912*log(2)/35 - 7153248/1225.
Sum_{n>=0} (-1)^n/a(n) = 73728*Pi/35 - 8105688/1225. (End)

A082985 Coefficient table for Chebyshev's U(2n,x) polynomial expanded in decreasing powers of (-4(1-x^2)).

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 7, 14, 7, 1, 9, 27, 30, 9, 1, 11, 44, 77, 55, 11, 1, 13, 65, 156, 182, 91, 13, 1, 15, 90, 275, 450, 378, 140, 15, 1, 17, 119, 442, 935, 1122, 714, 204, 17, 1, 19, 152, 665, 1729, 2717, 2508, 1254, 285, 19

Offset: 0

Gary W. Adamson, May 29 2003

Sum of row #n = A000204(2n+1), i.e., A002878(n).
Row #n has the unsigned coefficients of a polynomial whose roots are 2 sin(2*Pi*k/(2n+1)) [for k=1 to 2n].
The positive roots are the diagonal lengths of a regular (2n+1)-gon, inscribed in the unit circle.
Polynomial of row #n = Sum_{m=0..n} (-1)^m T(n,m) x^(2n-2m).
This is also the unsigned coefficient table of Chebyshev's 2*T(2*n+1,x) polynomials expanded in decreasing odd powers of 2*x. - Wolfdieter Lang, Mar 07 2007
The n-th row are the coefficients of the polynomial S(n) where S(0)=1, S(1)=x+3, and S(n) = (x+2)*S(n-1) - S(n-2) (see Sun link). - Michel Marcus, Mar 07 2016

			Expansion of polynomials:
  x^0;
  x^2  -  3*x^0;
  x^4  -  5*x^2 +  5*x^0;
  x^6  -  7*x^4 + 14*x^2 -  7*x^0;
  x^8  -  9*x^6 + 27*x^4 - 30*x^2 +  9*x^0;
  x^10 - 11*x^8 + 44*x^6 - 77*x^4 + 55*x^2 - 11*x^0; ...
Polynomial #4 has 8 roots: 2*sin(2*Pi*k/9) for k=1 to 8.
Coefficients (with signs removed) are
  1;
  1,  3;
  1,  5,  5;
  1,  7, 14,  7;
  1,  9, 27, 30,  9;
  1, 11, 44, 77, 55, 11;
  ...

J. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, 1992; see pp. 151,175.
Stephen Eberhart, "Mathematical-Physical Correspondence," Number 37-38, Jan 08, 1982.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Vincenzo Librandi, Rows n = 0..100 of triangle, flattened
K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681.
Zhi-Hong Sun, Expansions and identities concerning Lucas sequences, Fibonacci Quart. 44 (2006), no. 2, 145-153. See Theorem 3.1

Cf. companion triangle A084534.
Cf. diagonals: A005408, A000330, A005585, A050486, A054333, A057788.
Cf. A054142, A172431.

Flat(List([0..10], n-> List([0..n], k-> Binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1) ))); # G. C. Greubel, Dec 30 2019

[Binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 30 2019

A082985 := proc(n,m)
    binomial(2*n-m,m)*(2*n+1)/(2*n-2*m+1) ;
end proc: # R. J. Mathar, Sep 08 2013

T[n_, m_]:= Binomial[2*n-m, m]*(2*n+1)/(2*n-2*m+1); Table[T[n, m], {n, 0, 9}, {m, 0, n}]//Flatten (* Jean-François Alcover, Oct 08 2013, after R. J. Mathar *)

T(n,k)=binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1); \\ G. C. Greubel, Dec 30 2019

[[binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 30 2019

Triangle read by rows: row #n has n+1 terms. T(n,0)=1, T(n,n)=2n+1, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k, m-k).
T(k, s) = ((2k+1)/(2s+1))*binomial(k+s, 2s), 0 <= s <= k; then transpose the triangle. - Gary W. Adamson, May 29 2003
From Wolfdieter Lang, Mar 07 2007: (Start)
Signed version: a(n,m)=0 if n < m, otherwise a(n,m) = ((-1)^m)*binomial(2*n+1-m,m)*(2*n+1)/(2*n+1-m). From the Rivlin reference, p. 37, eq.(1.92), using the differential eq. for T(2*n+1,x). Also from Waring's formula.
Signed version: a(n,m)=0 if n < m, otherwise a(n,m) = ((-1)^m)*(Sum_{k=0..n-m} binomial(m+k,k)*binomial(2*n+1,2*(k+m)))/2^(2*(n-m)). Proof: De Moivre's formula for cos((2*n+1)*phi) rewritten in terms of odd powers of cos(phi). Cf. Rivlin reference p. 4, eq.(1.10).
Signed version: a(n,m) = A084930(n,n-m)/2^(2*(n-m)) (scaled coefficients of Chebyshev's T(2*n+1,x), decreasing odd powers).
Unsigned version: a(n,m)=0 if n < m, otherwise a(n,m) = binomial(2*n-m,m)*(2*n+1)/(2*(n-m)+1). From the differential eq. for U(2*n,x). (End)
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2). - Philippe Deléham, Feb 24 2012
Sum_{i>=0} T(n-i,n-2*i) = A003945(n). - Philippe Deléham, Feb 24 2012
Sum_{i>=0} T(n-i, n-2*i)*4^i = 3^n = A000244(n). - Philippe Deléham, Feb 24 2012
From Paul Weisenhorn Nov 25 2019: (Start)
T(r,k) = binomial(2*r-k,k-1) + binomial(2*r-1-k,k-2) with 1 <= r and 1 <= k <= r.
For a given n, one gets r = floor((1+sqrt(8*n))/2), k = n-(r^2-r)/2, a(n) = binomial(2*r-k,k-1) + binomial(2*r-1-k,k-2). (End)

Edited by Anne Donovan (anned3005(AT)aol.com), Jun 11 2003

Re-edited by Don Reble, Nov 12 2005

A110813 A triangle of pyramidal numbers.

Original entry on oeis.org

1, 3, 1, 5, 4, 1, 7, 9, 5, 1, 9, 16, 14, 6, 1, 11, 25, 30, 20, 7, 1, 13, 36, 55, 50, 27, 8, 1, 15, 49, 91, 105, 77, 35, 9, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1, 23, 121, 385, 825

Offset: 0

Paul Barry, Aug 05 2005

Triangle A029653 less first column. In general, the product (1/(1-x),x/(1-x))*(1+m*x,x) yields the Riordan array ((1+(m-1)x)/(1-x)^2,x/(1-x)) with general term T(n,k)=(m*n-(m-1)*k+1)*C(n+1,k+1)/(n+1). This is the reversal of the (1,m)-Pascal triangle, less its first column. - Paul Barry, Mar 01 2006
The column sequences give, for k=0..10: A005408 (odd numbers), A000290 (squares), A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.
Linked to Chebyshev polynomials by the fact that this triangle with interpolated zeros in the rows and columns is a scaled version of A053120.
Row sums are A033484. Diagonal sums are A001911(n+1) or F(n+4)-2. Factors as (1/(1-x),x/(1-x))*(1+2x,x). Inverse is A110814 or (-1)^(n-k)*A104709.
This triangle is a subtriangle of the [2,1] Pascal triangle A029653 (omit there the first column).
Subtriangle of triangles in A029653, A131084, A208510. - Philippe Deléham, Mar 02 2012
This is the iterated partial sums triangle of A005408 (odd numbers). Such iterated partial sums of arithmetic progression sequences have been considered by Narayana Pandit (see the Mar 20 2015 comment on A000580 where the MacTutor History of Mathematics archive link and the Gottwald et al. reference, p. 338, are given). - Wolfdieter Lang, Mar 23 2015

			The number triangle T(n, k) begins
n\k  0   1   2   3    4    5    6   7   8  9 10 11
0:   1
1:   3   1
2:   5   4   1
3:   7   9   5   1
4:   9  16  14   6    1
5:  11  25  30  20    7    1
6:  13  36  55  50   27    8    1
7:  15  49  91 105   77   35    9   1
8:  17  64 140 196  182  112   44  10   1
9:  19  81 204 336  378  294  156  54  11  1
10: 21 100 285 540  714  672  450 210  65 12  1
11: 23 121 385 825 1254 1386 1122 660 275 77 13  1
... reformatted by _Wolfdieter Lang_, Mar 23 2015
As a number square S(n, k) = T(n+k, k), rows begin
  1,   1,   1,   1,   1,   1, ...
  3,   4,   5,   6,   7,   8, ...
  5,   9,  14,  20,  27,  35, ...
  7,  16,  30,  50,  77, 112, ...
  9,  25,  55, 105, 182, 294, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000290, A000330, A002415, A005408, A005585, A029655, A040977, A050486, A053347, A054333, A054334, A057788.

Table[2*Binomial[n + 1, k + 1] - Binomial[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 19 2017 *)

for(n=0,10, for(k=0,n, print1(2*binomial(n+1, k+1) - binomial(n,k), ", "))) \\ G. C. Greubel, Oct 19 2017

Number triangle T(n, k) = C(n, k)*(2n-k+1)/(k+1) = 2*C(n+1, k+1) - C(n, k); Riordan array ((1+x)/(1-x)^2, x/(1-x)); As a number square read by antidiagonals, T(n, k)=C(n+k, k)(2n+k+1)/(k+1).
Equals A007318 * an infinite bidiagonal matrix with 1's in the main diagonal and 2's in the subdiagonal. - Gary W. Adamson, Dec 01 2007
Binomial transform of an infinite lower triangular matrix with all 1's in the main diagonal, all 2's in the subdiagonal and the rest zeros. - Gary W. Adamson, Dec 12 2007
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0)=T(1,1)=1, T(1,0)=3, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 30 2013
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(7 + 9*x + 5*x^2/2! + x^3/3!) = 7 + 16*x + 30*x^2/2! + 50*x^3/3! + 77*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014
T(n, k) = ps(1, 2; k, n-k) with ps(a, d; k, n) = sum(ps(a, d; k-1, j), j=0..n) and input ps(a, d; 0, j) = a + d*j. See the iterated partial sums comment from Mar 23 2015 above. - Wolfdieter Lang, Mar 23 2015
From Franck Maminirina Ramaharo, May 21 2018: (Start)
T(n,k) = coefficients in the expansion of ((x + 2)*(x + 1)^n - 2)/x.
T(n,k) = A135278(n,k) + A135278(n-1,k).
T(n,k) = A097207(n,n-k).
G.f.: (y + 1)/((y - 1)*(x*y + y - 1)).
E.g.f.: ((x + 2)*exp(x*y + y) - 2*exp(y))/x.
(End)

A007487 Sum of 9th powers.

Original entry on oeis.org

0, 1, 513, 20196, 282340, 2235465, 12313161, 52666768, 186884496, 574304985, 1574304985, 3932252676, 9092033028, 19696532401, 40357579185, 78800938560, 147520415296, 266108291793, 464467582161, 787155279940, 1299155279940, 2093435326521, 3300704544313

Offset: 0

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 815.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Bruno Berselli, A description of the recursive method in Formula lines (second formula): website Matem@ticamente (in Italian).
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Eric Weisstein's World of Mathematics, Power Sum.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Row 9 of array A103438.

Cf. A000217, A000542, A069094.

[&+[n^9: n in [0..m]]: m in [0..22]]; // Bruno Berselli, Aug 23 2011

[seq(add(i^9,i=1..n),n=0..40)];
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^9 od: seq(a[n], n=0..22); # Zerinvary Lajos, Feb 22 2008

lst={};s=0;Do[s=s+n^9;AppendTo[lst, s], {n, 10^2}];lst..or..Table[Sum[k^9, {k, 1, n}], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0,30]^9] (* Harvey P. Dale, Oct 09 2016 *)

a(n)=n^2*(n+1)^2*(n^2+n-1)*(2*n^4+4*n^3-n^2-3*n+3)/20 \\ Charles R Greathouse IV, Oct 07 2015

A007487_list, m = [0], [362880, -1451520, 2328480, -1905120, 834120, -186480, 18150, -510, 1, 0, 0]
for _ in range(10**2):
    for i in range(10):
        m[i+1]+= m[i]
    A007487_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

a(n) = n^2*(n+1)^2*(n^2+n-1)*(2*n^4+4*n^3-n^2-3*n+3)/20 (see MathWorld, Power Sum, formula 39). a(n) = n*A000542(n) - Sum_{i=0..n-1} A000542(i). - Bruno Berselli, Apr 26 2010
G.f.: x*(1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1-x)^11. a(n) = a(-n-1). - Bruno Berselli, Aug 23 2011
a(n) = -Sum_{j=1..9} j*Stirling1(n+1,n+1-j)*Stirling2(n+9-j,n). - Mircea Merca, Jan 25 2014
a(n) = (16/5)*A000217(n)^5 - 4*A000217(n)^4 + (12/5)*A000217(n)^3 - (3/5)*A000217(n)^2. - Michael Raney, Mar 14 2016
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n > 10. - Wesley Ivan Hurt, Dec 21 2016
a(n) = 288*A005585(n-1)^2 + 1728*A108679(n-3) + A062392(n)^2. - Yasser Arath Chavez Reyes, May 11 2024
a(n) = Sum_{i=1..n} J_9(i)*floor(n/i), where J_9 is A069094. - Ridouane Oudra, Jul 17 2025

A057788 Expansion of (1+x)/(1-x)^12.

Original entry on oeis.org

1, 13, 90, 442, 1729, 5733, 16744, 44200, 107406, 243542, 520676, 1058148, 2057510, 3848222, 6953544, 12183560, 20764055, 34512075, 56071470, 89224590, 139299615, 213696795, 322561200, 479634480, 703323660, 1018031196, 1455797448, 2058314440, 2879378332

Offset: 0

N. J. A. Sloane, Nov 04 2000

1/2^10 of twelfth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-12) is the number of 12-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
11-dimensional square numbers, tenth partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = sum_{i=0..n} C(n+10,i+10)*b(i), where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
2*a(n) is number of ways to place 10 queens on an (n+10) X (n+10) chessboard so that they diagonally attack each other exactly 45 times. The maximal possible attack number, p=binomial(k,2) =45 for k=10 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph. - Antal Pinter, Dec 27 2015

T. D. Noe, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
Index entries for linear recurrences with constant coefficients, signature (12,-66, 220,-495,792,-924,792,-495,220,-66,12,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A053120, A054334, A054333, A053347, A002415, A005585, A040977, A050486.
Partial sums of A054334.
Sixth column of A111125 (s=5, without leading zeros). - Wolfdieter Lang, Oct 18 2012

List([0..30], n -> (2*n+11)*Binomial(n+10, 10)/11); # G. C. Greubel, Dec 02 2018

[Binomial(n+10,10)*(2*n+11)/11: n in [0..40]]; // Vincenzo Librandi, Feb 14 2016

A057788 := proc(n)
        1/39916800*(2*n+11) *(n+10) *(n+9) *(n+8) *(n+7) *(n+6) *(n+5) *(n+4) *(n+3) *(n+2) *(n+ 1) ; end proc: # R. J. Mathar, Mar 22 2011

Table[(2*n+11)*Binomial[n+10, 10]/11, {n,0,40}] (* G. C. Greubel, Dec 02 2018 *)
CoefficientList[Series[(1 + x) / (1 - x)^12, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 14 2016 *)
LinearRecurrence[{12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{1,13,90,442,1729,5733,16744,44200,107406,243542,520676,1058148},30] (* Harvey P. Dale, Sep 07 2022 *)

Vec((1+x)/(1-x)^12+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

[(2*n+11)*binomial(n+10, 10)/11 for n in range(40)] # G. C. Greubel, Dec 02 2018

a(n) = 2*C(n+11, 11) - C(n+10, 10). - Paul Barry, Mar 04 2003
a(n) = C(n+10,10) + 2*C(n+10,11). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
a(n) = C(n+10,10)*(2n+11)/11. - Antal Pinter, Dec 27 2015
a(n) = 12*a(n-1)-66*a(n-2)+220*a(n-3)-495*a(n-4)+792*a(n-5)-924*a(n-6)+792*a(n-7)-495*a(n-8)+220*a(n-9)-66*a(n-10)+12*a(n-11)-a(n-12) for n >11. - Vincenzo Librandi, Feb 14 2016
a(n) = (2*n+11)*binomial(n+10, 10)/11. - G. C. Greubel, Dec 02 2018
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=0} 1/a(n) = 419751541/13230 - 2883584*log(2)/63.
Sum_{n>=0} (-1)^n/a(n) = 720896*Pi/63 - 237793798/6615. (End)

A107600 Column 5 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

1, 18, 101, 357, 978, 2274, 4711, 8954, 15915, 26806, 43197, 67079, 100932, 147798, 211359, 296020, 406997, 550410, 733381, 964137, 1252118, 1608090, 2044263, 2574414, 3214015, 3980366, 4892733, 5972491, 7243272, 8731118, 10464639

Offset: 9

Alford Arnold, May 17 2005

The sequences in A089574 count ordered partitions. Sequence A001296 can be associated with 9 = 3+3+3. Six times sequence A005585, associated with 10 = 3+3+2+2. The other three sequences comprising A107600 are generated in A034261 and can be associated with 10 = 5 + 5 = 4 + 4 + 2 = 2 + 2 + 2 + 2 + 2.

			A107600(n) can be constructed from five other sequences as follows:
1...7...25...65...140.......A001296
....1...11...56...196.......A034264
....6...42..162...462.......6.*.A005585.
....3...18...60...150.......A006011
....1....5...14....30.......A000330
therefore
1..18..101..357...978.......A107600

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A034261, A089574, A000330, A001296, A005585, A006011, A034264.

a:= n-> `if` (n<9, 0, (92292 +(-6580 +(-5745 +(1535 +(-147+5*n) *n) *n) *n) *n) *n /720 -218): seq(a(n), n=9..45); # Alois P. Heinz, Nov 06 2009

Select[CoefficientList[Series[(x^5-5x^4+7x^3+4x^2-11x-1)x^9/(x-1)^7, {x,0,50}],x],#>0&] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1,18,101,357,978,2274,4711},42] (* Harvey P. Dale, May 01 2011 *)

G.f.: (x^5 -5*x^4 +7*x^3 +4*x^2 -11*x -1) *x^9 /(x-1)^7. - Alois P. Heinz, Nov 06 2009

More terms from Alois P. Heinz, Nov 06 2009

A084930 Triangle of coefficients of Chebyshev polynomials T_{2n+1} (x).

Original entry on oeis.org

1, -3, 4, 5, -20, 16, -7, 56, -112, 64, 9, -120, 432, -576, 256, -11, 220, -1232, 2816, -2816, 1024, 13, -364, 2912, -9984, 16640, -13312, 4096, -15, 560, -6048, 28800, -70400, 92160, -61440, 16384, 17, -816, 11424, -71808, 239360, -452608, 487424, -278528, 65536, -19, 1140, -20064, 160512, -695552

Offset: 0

Gary W. Adamson, Jun 12 2003

From Herb Conn, Jan 28 2005: (Start)
"Letting x = 2 Cos 2A, we have the following trigonometric identities:
"Sin 3A = 3*Sin A - 4*Sin^3 A
"Sin 5A = 5*Sin A - 20*Sin^3 A + 16*Sin^5 A
"Sin 7A = 7*Sin A - 56*Sin^3 A + 112*Sin^5 A - 64*Sin^7 A
"Sin 9A = 9*Sin A - 120*Sin^3 A + 432*Sin^5 A - 576*Sin^7 A + 256*Sin^9 A, etc." (End)
Cayley (1876) states "Write sin u = x, then we have  sin u = x, [...] sin 3u = 3x - 4x^3, [...] sin 5u = 5x - 20x^3 + 16 x^5, [...]". Since T_n(cos(u)) = cos(nu) for all integer n, sin(u) = cos(u - Pi/2), and sin(u + k*Pi) = (-1)^k sin(u) it follows that T_n(sin(u)) = (-1)^((n-1)/2) sin(nu) for all odd integer n. - Michael Somos, Jun 19 2012
From Wolfdieter Lang, Aug 05 2014: (Start)
The coefficient triangle t(n,k) for the row polynomials Todd(n, x) := T_{2*n+1}(sqrt(x))/sqrt(x) = sum(t(n,k)*x^k, k=0..n) is the Riordan triangle ((1-z)/(1+z)^2, 4*z/(1+z)^2) (rewrite the g.f. for the present triangle a(n,k) given in the formula section). The triangle entries t(n,k) = a(n,k), but the interpretation of the row polynomials is different for both cases.
From the relation Todd(n, x) = S(n, 2*(2*x-1)) - S(n-1, 2*(2*x-1)) with the Chebyshev S-polynomials (see A049310 and the formula section of A130777) follows the recurrence: Todd(n, x) = 2*(-1)^n*(1-x)*Todd(n-1, 1-x) + (2x-1)*Todd(n-1, x), n >= 1, Todd(0, x) = 1.
This corresponds to the triangle recurrence t(n,k) = (2*(k+1)*(-1)^(n-k) - 1)*t(n-1,k) + 2*(1 +(-1)^(n-k))*t(n-1,k-1) + 2*(-1)^(n-k)*sum(binomial(l+1,k)*t(n-1,l), l=k+1..n-1), n >= k >= 1, t(n,k) = 0 if n < k, t(n,0) = (-1)^n*(2*n+1). Compare this with the shorter recurrence involving the rational A-sequence for this Riordan triangle which has g.f. x^2/(2-x-2*sqrt(1-x)). t(n,k) = sum(A(j)*t(n-1,k-1+j), j=0..n-k), n >= k >= 1. The Z-sequence has g.f. -(1 + 2/sqrt(1-x)). For the A- and Z-sequence see a link under A006232. (End)

			The triangle a(n,k):
n   2n+1\k 0     1      2       3       4        5        6         7        8        9      10 ...
0    1:    1
1    3:   -3     4
2    5:    5   -20     16
3    7:   -7    56   -112      64
4    9:    9  -120    432    -576     256
5   11:  -11   220  -1232    2816   -2816     1024
6   13:   13  -364   2912   -9984   16640   -13312     4096
7   15:  -15   560  -6048   28800  -70400    92160   -61440     16384
8   17:   17  -816  11424  -71808  239360  -452608   487424   -278528    65536
9   19:  -19  1140 -20064  160512 -695552  1770496 -2723840   2490368 -1245184   262144
10  21:   21 -1540  33264 -329472 1793792 -5870592 12042240 -15597568 12386304 -5505024 1048576
... formatted and extended by _Wolfdieter Lang_, Aug 02 2014.
---------------------------------------------------------------------------------------------------
First few polynomials T_{2n+1}(x) are
1*x - 3*x + 4*x^3
5*x - 20*x^3 + 16*x^5
- 7*x + 56*x^3 - 112*x^5 + 64*x^7
9*x - 120*x^3 + 432*x^5 - 576*x^7  + 256*x^9

A. Cayley, On an Expression for 1 +- sin(2p+1)u in Terms of sin u, Messenger of Mathematics, 5 (1876), pp. 7-8 = Mathematical Papers Vol. 10, n. 630, pp. 1-2.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2nd ed., Wiley, New York, 1990. p. 37, eq. (1.96) and p. 4, eq. (1.10).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.
Index entries for sequences related to Chebyshev polynomials.

Cf. A002315, A028297.
Cf. A053120 (coefficient triangle of T-polynomials), A127674 (even-indexed T polynomials).
Cf. A127675 (row reversed triangle with different signs).
Cf. A006232, A008310, A049310, A130777.
Cf. A157142, A000330(n), A005585, A050486, A054333.

row[n_] := (T = ChebyshevT[2*n+1, x]; Coefficient[T, x, #]& /@ Range[1, Exponent[T, x], 2]); Table[row[n], {n, 0, 9} ] // Flatten (* Jean-François Alcover, Aug 06 2014 *)

Alternate rows of A008310.
a(n,k)=((-1)^(n-k))*(2^(2*k))*binomial(n+1+k,2*k+1)*(2*n+1)/(n+1+k) if n>=k>=0 else 0.
From Wolfdieter Lang, Aug 02 2014: (Start)
a(n,k) = [x^(2*k+1)] T_{2*n+1}(x), n >= k >= 0.
G.f. for row polynomials: x*(1-z)/(1 + 2*(1- 2*x^2)*z + z^2). (End)
The first column sequences are: A157142, 4*(-1)^(n+1)*A000330(n), 16*(-1)^n*A005585(n-1), 64*(-1)^(n+1)*A050486(n-3), 256*(-1)^n*A054333(n-4), ... - Wolfdieter Lang, Aug 05 2014

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 26 2003

Edited; example rewritten (to conform with the triangle) by Wolfdieter Lang, Aug 02 2014

A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.

Original entry on oeis.org

1, 12, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original name: The first summation of row 4 of Euler's triangle - a row that will recursively accumulate to the power of 4.

D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883
Eric Weisstein, Link to section of MathWorld: Eulerian Number
Eric Weisstein, Link to section of MathWorld: Nexus number
Eric Weisstein, Link to section of MathWorld: Finite Differences
Index entries for linear recurrences with constant coefficients, signature (1).

For other sequences based upon MagicNKZ(n,k,z):
..... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7
---------------------------------------------------------------------------
z = 0 | A000007 | A019590 | .......MagicNKZ(n,k,0) = A008292(n,k+1) .......
z = 1 | A000012 | A040000 | A101101 | thisSeq | A101100 | ....... | .......
z = 2 | A000027 | A005408 | A008458 | A101103 | A101095 | ....... | .......
z = 3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | .......
z = 4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | .......
z = 5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181
z = 6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177
z = 7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523
z = 8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015
z = 9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541
Cf. A101095 for an expanded table and more about MagicNKZ.

MagicNKZ = Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 4, 4}, {z, 1, 1}, {k, 0, 34}]
Join[{1, 12, 23},LinearRecurrence[{1},{24},56]] (* Ray Chandler, Sep 23 2015 *)

a(k) = MagicNKZ(4,k,1) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n (cf. A101095). That is, a(k) = Sum_{j=0..k+1} (-1)^j*binomial(4, j)*(k-j+1)^4.
a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4. - Joerg Arndt, Nov 30 2014
G.f.: x*(1+11*x+11*x^2+x^3)/(1-x). - Colin Barker, Apr 16 2012

New name from Joerg Arndt, Nov 30 2014

Original Formula edited and Crossrefs table added by Danny Rorabaugh, Apr 22 2015

A108679 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)^2(n+5)^2*(n+6)/86400.

Original entry on oeis.org

1, 21, 196, 1176, 5292, 19404, 60984, 169884, 429429, 1002001, 2186184, 4504864, 8836464, 16604784, 30046752, 52581816, 89311761, 147685461, 238369516, 376372920, 582481900, 885069900, 1322357400, 1945206900, 2820550005, 4035556161, 5702666256, 7965629056

Offset: 0

Emeric Deutsch, Jun 17 2005

Kekulé numbers for certain benzenoids.
6th column of the table of Narayana numbers A001263. - Zerinvary Lajos, Jun 18 2007
Sequence provided by binomial(n-1,m)*binomial(n,m)/(m+1) for m=5 and n>5 (these numbers are also called Runyon numbers, see T. Koshy in References). - Vincenzo Librandi, Sep 04 2014

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 232, # 1).
T. Koshy, Catalan Numbers with Applications, Oxford University Press, 2009, p. 7.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; Prop. 8.4, case n=7. - N. J. A. Sloane, Aug 28 2010

T. D. Noe, Table of n, a(n) for n = 0..1000
Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 25.

Cf. A001263, A002378, A005585, A006542, A006857 (sequences having a similar structure), A288876.

[Binomial(n-1,5)*Binomial(n,5)/6: n in [6..45]]; // Vincenzo Librandi, Sep 04 2014

a:=n->(n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)^2*(n+6)/86400: seq(a(n),n=0..40);

Table[(n + 1) (n + 2)^2 (n + 3)^2 (n + 4)^2 (n + 5)^2 (n + 6)/86400,{n, 0, 50}]  (* Harvey P. Dale, Mar 13 2011 *)

Vec((1+10*x+20*x^2+10*x^3+x^4)/(1-x)^11 + O(x^99)) \\ Altug Alkan, Sep 02 2016

def A108679(n): return binomial(n+5,5)*binomial(n+6,5)//6
print([A108679(n) for n in range(41)]) # G. C. Greubel, Mar 12 2025

a(n) = binomial(n+5, 5)*binomial(n+6, 5)/6 = binomial(n+6, 6)*binomial(n+6, 5)/(n+6).
a(n) = A001263(n+6,6).
G.f.: (1 + 10*x + 20*x^2 + 10*x^3 + x^4)/(1 - x)^11. Numerator polynomial is the fifth row polynomial of the Narayana triangle.
a(n) = binomial(n+5,5)^2 - binomial(n+5,4)*binomial(n+5,6). - Gary Detlefs, Dec 05 2011
a(n) = Product_{i=1..5} A002378(n+i)/A002378(i). - Bruno Berselli, Sep 01 2016
From Amiram Eldar, Oct 19 2020: (Start)
Sum_{n>=0} 1/a(n) = 27637/2 - 1400*Pi^2.
Sum_{n>=0} (-1)^n/a(n) = 2560*log(2) - 3547/2. (End)
a(n) = (A005585(n+2)^2 - A288876(n+1))/24. - Yasser Arath Chavez Reyes, Aug 19 2024

cp's OEIS Frontend

A054334 1/512 of 11th unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).

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A054333 1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).

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A082985 Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).

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A110813 A triangle of pyramidal numbers.

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A007487 Sum of 9th powers.

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A057788 Expansion of (1+x)/(1-x)^12.

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A082985 Coefficient table for Chebyshev's U(2n,x) polynomial expanded in decreasing powers of (-4(1-x^2)).

A108679 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)^2(n+5)^2*(n+6)/86400.