A053120 -id:A053120 - 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)

A006974 Coefficients of Chebyshev T polynomials: a(n) = A053120(n+8, n), n >= 0.

Original entry on oeis.org

1, 9, 50, 220, 840, 2912, 9408, 28800, 84480, 239360, 658944, 1770496, 4659200, 12042240, 30638080, 76873728, 190513152, 466944000, 1133117440, 2724986880, 6499598336, 15386804224, 36175872000, 84515225600, 196293427200, 453437816832

Offset: 0

Simon Plouffe

If X_1,X_2,...,X_n are 2-blocks of a (2n+1)-set X then, for n>=3, a(n-3) is the number of (n+4)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007

The fourth corrector line for transforming 2^n offset 0 with a leading 1 into the Fibonacci sequence. [Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009]

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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
M. H. Albert, M. D. Atkinson, R. Brignall, The enumeration of three pattern classes using monotone grid classes, El. J. Combinat. 19 (3) (2012) P20, Chapter 5.4.1.
Index entries for sequences related to Chebyshev polynomials.

Cf. A039991 (see column 8), A003472 (partial sums), A053120.

[2^(n-1)/4*Binomial(n+3,3)*(n+8) : n in [0..25]]; // Brad Clardy, Mar 08 2012

Maple
```
a := n->n*(n+1)*(n+2)*(n+7)*2^(n-5)/3;
```

G.f.: (1-x)/(1-2*x)^5.
a(n) = Sum_{k=0..floor((n+8)/2)} C(n+8, 2k)*C(k, 4). - Paul Barry, May 15 2003
Binomial transform of a(n)=(24*n^4-134*n^3+261*n^2-130*n+3)/3 offset 0. a(3)=220. [Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009]
a(n) = 2^(n-3)*binomial(n+3, 3)*(n+8). - Brad Clardy, Mar 08 2012 [See a comment in A053120 on subdiagonals. - Wolfdieter Lang, Jan 03 2020]
E.g.f.: (1/3)*exp(2*x)*(3 + 21*x + 27*x^2 + 10*x^3 + x^4). - Stefano Spezia, Aug 17 2019

Name clarified by Wolfdieter Lang, Nov 26 2019

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)

A006975 Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+10, n), n >= 0.

Original entry on oeis.org

1, 11, 72, 364, 1568, 6048, 21504, 71808, 228096, 695552, 2050048, 5870592, 16400384, 44843008, 120324096, 317521920, 825556992, 2118057984, 5369233408, 13463453696, 33426505728, 82239815680, 200655503360, 485826232320

Offset: 0

Simon Plouffe

Binomial transform of A069038. - Paul Barry, Feb 19 2003
If X_1, X_2, ..., X_n are 2-blocks of a (2n+1)-set X then, for n>=4, a(n-4) is the number of (n+5)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
The 5th corrector line for transforming 2^n offset 0 with a leading 1 into the Fibonacci sequence. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paolo Xausa, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
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].
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

First differences of A054849.

Cf. A039991, A053120, A069038.

[2^(n-1)/5*Binomial(n+4,4)*(n+10): n in [0..25]]; // Brad Clardy, Mar 10 2012

Table[2^(n-1)/5*Binomial[n+4, 4]*(n+10), {n, 0, 30}] (* Paolo Xausa, Jun 26 2024 *)

G.f.: (1-x)/(1-2*x)^6. a(n) = 2^(n-1)*binomial(n+4, 4)*(n+10)/5, for n >= 0. [a(n) from Mar 06 2000 rewritten. See the Brad Clardy formula below, and a comment in A053120 on subdiagonals. - Wolfdieter Lang, Jan 03 2020]
a(n) = 2^(n-4)*(n+1)(n+2)(n+3)(n+4)(n+10)/15. - Paul Barry, Feb 19 2003
a(n) = Sum_{k=0..floor((n+10)/2)} C(n+10, 2k)*C(k, 5). - Paul Barry, May 15 2003
a(n) = -A039991(n+10, 10). - N. J. A. Sloane, May 16 2003
a(n) = binomial transform of b(n)= (2*n^5 + 10*n^4 + 30*n^3 + 50*n^2 + 43*n + 15) / 15 offset 0. a(3) = 364. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009
a(n) = 2^(n-1)/5*binomial(n+4,4)*(n+10). - Brad Clardy, Mar 10 2012
E.g.f.: (1/15)*exp(2*x)*(15+135*x+240*x^2+140*x^3+30*x^4+2*x^5). - Stefano Spezia, Jan 03 2020

Name clarified by Wolfdieter Lang, Nov 26 2019

A006976 Coefficients of Chebyshev T polynomials: a(n) = A053120(n+12, n), n >= 0.

Original entry on oeis.org

1, 13, 98, 560, 2688, 11424, 44352, 160512, 549120, 1793792, 5637632, 17145856, 50692096, 146227200, 412778496, 1143078912, 3111714816, 8341487616, 22052208640, 57567870976, 148562247680, 379364311040, 959384125440

Offset: 0

Simon Plouffe

Binomial transform of A069039. - Paul Barry, Feb 19 2003

If X_1, X_2, ..., X_n are 2-blocks of a (2n+1)-set X then, for n >= 5, a(n-5) is the number of (n+6)-subsets of X intersecting each X_i, (i = 1, 2, ..., n). - Milan Janjic, Nov 18 2007

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, 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
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

a(n) = A039991(n+12, 12), A053120.

Partial sums are in A002409.

List([0..25], n-> 2^(n-1)*Binomial(n+5,5)*(n+12)/6); # G. C. Greubel, Aug 27 2019

[2^(n-1)/6*Binomial(n+5,5)*(n+12) : n in [0..25]]; // Brad Clardy, Mar 10 2012

seq(2^(n-1)*binomial(n+5,5)*(n+12)/6, n=0..25); # G. C. Greubel, Aug 27 2019

Table[2^(n-1)*Binomial[n+5,5]*(n+12)/6, {n,0,25}] (* G. C. Greubel, Aug 27 2019 *)
LinearRecurrence[{14,-84,280,-560,672,-448,128},{1,13,98,560,2688,11424,44352},30] (* Harvey P. Dale, Sep 26 2024 *)

vector(26, n, 2^(n-2)*binomial(n+4,5)*(n+11)/6) \\ G. C. Greubel, Aug 27 2019

[2^(n-1)*binomial(n+5,5)*(n+12)/6 for n in (0..25)] # G. C. Greubel, Aug 27 2019

G.f.: (1-x)/(1-2*x)^7.
a(n) = 2^n*binomial(n+5, 5)*(n+12)/12. [See a comment in A053120 on subdiagonal sequences. - Wolfdieter Lang, Jan 03 2020]
a(n) = Sum_{k = 0..floor((n+12)/2)} C(n+12,2*k)*C(k,6). - Paul Barry, May 15 2003
E.g.f.: (1/45)*exp(2*x)*(45 + 495*x + 1125*x^2 + 900*x^3 + 300*x^4 + 42*x^5 + 2*x^6). - Stefano Spezia, Jan 03 2020

More terms from James Sellers, Aug 21 2000

Name clarified by Wolfdieter Lang, Nov 26 2019

A209404 Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+14, n), n >= 0.

Original entry on oeis.org

1, 15, 128, 816, 4320, 20064, 84480, 329472, 1208064, 4209920, 14057472, 45260800, 141213696, 428654592, 1270087680, 3683254272, 10478223360, 29297934336, 80648077312, 218864025600, 586290298880, 1551944908800, 4063273943040

Offset: 0

Brad Clardy, Mar 08 2012

The MAGMA program provided calculates the coefficients of order one Chebyshev polynomials, for any arbitrary level. For example, setting Rn to 0 produces A001792, 1 produces A001793, 2 produces A001794, 3 produces A006974, 4 produces A006975, and 5 produces A006976. This sequence is produced with an Rn of 6.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Cf. A001792, A001793, A001794, A006974, A006975, A006976, A053120.

List([0..30], n-> 2^(n-1)*(n+14)*Binomial(n+6,6)/7); # G. C. Greubel, Oct 18 2019

Rn:=6; [2^(n-1)/(Rn+1)*Binomial(n+Rn,Rn)*(n+(Rn+1)*2) : n in [0..22]];

R:=PowerSeriesRing(Integers(), 23); Coefficients(R!( (1-x)/(1-2*x)^8 )); // Marius A. Burtea, Oct 17 2019

seq(2^(n-1)*(n+14)*binomial(n+6,6)/7, n=0..30); # G. C. Greubel, Oct 18 2019

CoefficientList[Series[(1-x)/(1-2*x)^8, {x,0,30}], x] (* or *) Table[2^(n-1)*Binomial[n+6,6]*(n+14)/7, {n,0,30}] (* G. C. Greubel, Jan 03 2018 *)

for(n=0,30, print1(2^(n-1)*binomial(n+6,6)*(n+14)/7, ", ")) \\ G. C. Greubel, Jan 03 2018

a(n) = 2^(n-1)*binomial(n+6, 6)*(n+14)/7 = -A053120(n+14, n), n >= 0. [See a comment in A053120 on subdiagonal sequences. - Wolfdieter Lang, Jan 03 2020]
G.f.: (1-x)/(1-2*x)^8. - Colin Barker, May 31 2013
E.g.f.: (1/315)*exp(2*x)*(315 + 4095*x + 11340*x^2 + 11550*x^3 + 5250*x^4 + 1134*x^5 + 112*x^6 + 4*x^7). - Stefano Spezia, Oct 17 2019

Name made more specific by Wolfdieter Lang, Nov 25 2019

A111526 Number triangle T(n,k)=C((n+k)/2,k)(n+1)(1+(-1)^(n-k))/(2(k+1)); T(n,k)=(-1)^((n-k)/2)*A053120(n+1,k+1)/2^k; Riordan array ((1+x^2)/(1-x^2)^2,x/(1-x^2)).

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 0, 4, 0, 1, 5, 0, 5, 0, 1, 0, 9, 0, 6, 0, 1, 7, 0, 14, 0, 7, 0, 1, 0, 16, 0, 20, 0, 8, 0, 1, 9, 0, 30, 0, 27, 0, 9, 0, 1, 0, 25, 0, 50, 0, 35, 0, 10, 0, 1, 11, 0, 55, 0, 77, 0, 44, 0, 11, 0, 1, 0, 36, 0, 105, 0, 112, 0, 54, 0, 12, 0, 1, 13, 0, 91, 0, 182, 0, 156, 0, 65, 0, 13, 0, 1, 0

Offset: 0

Paul Barry, Aug 05 2005

A scaled Chebyshev triangle.

Row sums are A001350(n+1). Diagonal sums are A033484, with interpolated zeros. Inverse is A111527.

			Triangle starts
1;
0,1;
3,0,1;
0,4,0,1;
5,0,5,0,1;
0,9,0,6,0,1;
7,0,14,0,7,0,1;

Cf. A110813.

A136265 Integral form of A053120 :Triangle of coefficients of Integral form Chebyshev's T(n, x) polynomials (powers of x in increasing order); Much improved version by use of the integro-differential recursive form over a previous attempt.

Original entry on oeis.org

1, -1, 2, 0, -4, 2, 3, -2, -12, 4, 0, 16, -6, -32, 8, -5, 2, 60, -16, -80, 16, 0, -36, 10, 192, -40, -192, 32, 7, -2, -168, 36, 560, -96, -448, 64, 0, 64, -14, -640, 112, 1536, -224, -1024, 128, -9, 2, 360, -64, -2160, 320, 4032, -512, -2304, 256, 0, -100, 18, 1600, -240, -6720, 864, 10240, -1152, -5120, 512, 11, -2

Offset: 1

Roger L. Bagula, Mar 18 2008

Row sums are:
Join[{1}, Table[Apply[Plus, CoefficientList[2*x*P[x, n] - Q[x, n], x]], {n,
0, 10}]];
{1, 1, -2, -7, -14, -23, -34, -47, -62, -79, -98, -119}
Integration of the doubled functions is not orthogonal:
Table[Table[Integrate[Sqrt[1/(1 - x^2)]*(2*x*P[x, n] - Q[x, n])*(2*x*P[x, m] -
Q[x, m]), {x, -1, 1}], {n, 0, 10}], {m, 0, 10}]

			{1},
{-1, 2},
{0, -4, 2},
{3, -2, -12, 4},
{0, 16, -6, -32, 8},
{-5,2, 60, -16, -80, 16},
{0, -36, 10, 192, -40, -192, 32},
{7, -2, -168, 36, 560, -96, -448, 64},
{0, 64, -14, -640, 112, 1536, -224, -1024, 128},
{-9, 2, 360, -64, -2160, 320, 4032, -512, -2304, 256},
{0, -100, 18, 1600, -240, -6720, 864, 10240, -1152, -5120, 512},
{11, -2, -660, 100,6160, -800, -19712, 2240, 25344, -2560, -11264, 1024}

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986,Pages 42-50

Cf. A053120.

P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; Q[x_, n_] := D[P[x, n + 1], x]; Table[ExpandAll[2*x*P[x, n] - Q[x, n]], {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[2*x*P[x, n] - Q[x, n], x], {n, 0, 10}]]; Join[{1}, Table[Apply[Plus, CoefficientList[2*x*P[x, n] - Q[x, n], x]], {n, 0, 10}]]; Flatten[a]

P(x, n) = 2*x*P(x, n - 1) - P(x, n - 2); Q(x, n) := D[P[x, n + 1], x]=dp(x,n)/dx Output Integral form: IP(x,n)=2*x*p(x,n)-Q(x,n)

A136523 Triangle T(n,k) = A053120(n,k) + A053120(n-1,k), read by rows.

Original entry on oeis.org

1, 1, 1, -1, 1, 2, -1, -3, 2, 4, 1, -3, -8, 4, 8, 1, 5, -8, -20, 8, 16, -1, 5, 18, -20, -48, 16, 32, -1, -7, 18, 56, -48, -112, 32, 64, 1, -7, -32, 56, 160, -112, -256, 64, 128, 1, 9, -32, -120, 160, 432, -256, -576, 128, 256, -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512

Offset: 0

Roger L. Bagula, Mar 23 2008

			Triangle begins as:
   1;
   1,  1;
  -1,  1,   2;
  -1, -3,   2,    4;
   1, -3,  -8,    4,    8;
   1,  5,  -8,  -20,    8,   16;
  -1,  5,  18,  -20,  -48,   16,   32;
  -1, -7,  18,   56,  -48, -112,   32,   64;
   1, -7, -32,   56,  160, -112, -256,   64,   128;
   1,  9, -32, -120,  160,  432, -256, -576,   128, 256;
  -1,  9,  50, -120, -400,  432, 1120, -576, -1280, 256, 512;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A001792, A001793, A001794, A000330, A008794, A011782.
Cf. A025192, A040000 (row sums), A053120, A057077, A081277, A109613.
Cf. A124182.

function A053120(n,k)
  if ((n+k) mod 2) eq 1 then return 0;
  elif n eq 0 then return 1;
  else return (-1)^Floor((n-k)/2)*(n/(n+k))*Binomial(Floor((n+k)/2), k)*2^k;
  end if;
end function;
A136523:= func< n,k | A053120(n,k) + A053120(n-1,k) >;
[A136523(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 26 2023

A053120[n_, k_]:= Coefficient[ChebyshevT[n,x], x, k];
T[n_, k_]:= T[n, k]= A053120[n,k] + A053120[n-1,k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A053120(n,k):
    if (n+k)%2==1: return 0
    elif n==0: return 1
    else: return floor((-1)^((n-k)//2)*(n/(n+k))*binomial((n+k)//2, k)*2^k)
def A136523(n,k): return A053120(n,k) + A053120(n-1,k)
flatten([[A136523(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 26 2023

T(n, k) = A053120(n,k) + A053120(n-1,k).
Sum_{k=0..n} T(n, k) = A040000(n).
From G. C. Greubel, Jul 26 2023: (Start)
T(n, 0) = A057077(n).
T(n, 1) = (-1)^floor((n-1)/2) * A109613(n-1).
T(n, 2) = (-1)^floor((n-2)/2) * A008794(n-1).
T(n, 3) = (-1)^floor((n+1)/2) * A000330(n-1).
T(n, n) = A011782(n).
T(n, n-1) = A011782(n-1).
T(n, n-2) = -A001792(n-2).
T(n, n-4) = A001793(n-3).
T(n, n-6) = -A001794(n-6).
Sum_{k=0..n} (-1)^k*T(n,k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n) + [n=1].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^floor(n/2)*A025192(floor(n/2)). (End)

Edited by G. C. Greubel, Jul 26 2023

A136663 Triangle of coefficients of the Pascal sum of A053120 Chebyshev's T(n, x) polynomials :p(x,n)=2xp(x,n-1)-p(x,n-2); pp(x,n)=Sum[Binomial[n,m]*p(x,m),{m,0,n}].

Original entry on oeis.org

1, 1, 1, 0, 2, 2, -2, 0, 6, 4, -4, -8, 4, 16, 8, -4, -20, -20, 20, 40, 16, 0, -24, -72, -40, 72, 96, 32, 8, 0, -112, -224, -56, 224, 224, 64, 16, 64, -32, -448, -624, 0, 640, 512, 128, 16, 144, 288, -288, -1584, -1584, 384, 1728, 1152, 256, 0, 160, 800, 960, -1600, -5088, -3680, 1920, 4480, 2560, 512

Offset: 1

Roger L. Bagula, Apr 02 2008

Row sums:

{1, 0, -2, -6, -14, -30, -62, -126, -254, -510, -1022}

			{1},
{1, 1},
{0, 2, 2},
{-2, 0, 6, 4},
{-4, -8, 4, 16, 8},
{-4, -20, -20, 20, 40, 16},
{0, -24, -72, -40, 72, 96, 32},
{8, 0, -112, -224, -56, 224, 224, 64},
{16, 64, -32, -448, -624, 0, 640, 512, 128},
{16, 144, 288, -288, -1584, -1584, 384, 1728, 1152, 256},
{0,160, 800, 960, -1600, -5088, -3680, 1920, 4480, 2560, 512}

Cf. A053120.

P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; Q[x_, n_] := Q[x, n] = Sum[P[x, m]*Binomial[n, m], {m, 0, n}]; a = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a]

p(x,n)=2*x*p(x,n-1)-p(x,n-2); pp(x,n)=Sum[Binomial[n,m]*p(x,m),{m,0,n}]

cp's OEIS Frontend

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

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A006974 Coefficients of Chebyshev T polynomials: a(n) = A053120(n+8, n), n >= 0.

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

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A006975 Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+10, n), n >= 0.

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A006976 Coefficients of Chebyshev T polynomials: a(n) = A053120(n+12, n), n >= 0.

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A209404 Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+14, n), n >= 0.

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A136663 Triangle of coefficients of the Pascal sum of A053120 Chebyshev's T(n, x) polynomials :p(x,n)=2xp(x,n-1)-p(x,n-2); pp(x,n)=Sum[Binomial[n,m]*p(x,m),{m,0,n}].