A001794 -id:A001794 - cp's OEIS frontend

1, 5, 1, 19, 6, 1, 63, 25, 7, 1, 192, 88, 32, 8, 1, 552, 280, 120, 40, 9, 1, 1520, 832, 400, 160, 49, 10, 1, 4048, 2352, 1232, 560, 209, 59, 11, 1, 10496, 6400, 3584, 1792, 769, 268, 70, 12, 1, 26624, 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1, 66304, 43520

Offset: 0

Wolfdieter Lang, May 26 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is (((1-z)^3)/(1-2*z)^4)/(1-x*z/(1-z)).
This is the fourth member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.
The column sequences appear as A049612(n+1), A055585, A001794, A001789(n+3), A027608, A055586 for m=0..5.

			[0] 1
[1] 5, 1
[2] 19, 6, 1
[3] 63, 25, 7, 1
[4] 192, 88, 32, 8, 1
[5] 552, 280, 120, 40, 9, 1
[6] 1520, 832, 400, 160, 49, 10, 1
[7] 4048, 2352, 1232, 560, 209, 59, 11, 1
Fourth row polynomial (n=3): p(3, x)= 63 + 25*x + 7*x^2 + x^3.

Cf. A007318, A055248, A055249, A055252. Row sums: A049600(n+1, 4).

T := (n, k) -> binomial(n, k)*hypergeom([4, k - n], [k + 1], -1):
for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # Peter Luschny, Sep 23 2024

a(n, m)=sum(A055252(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j, m), j=m..n-1)+ A055252(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: (((1-x)^3)/(1-2*x)^4)*(x/(1-x))^m, m >= 0.

T(n, k) = binomial(n, k)*hypergeom([4, k - n], [k + 1], -1). - Peter Luschny, Sep 23 2024

A207537 Triangle of coefficients of polynomials u(n,x) jointly generated with A207538; see Formula section.

Original entry on oeis.org

1, 2, 1, 4, 3, 8, 8, 1, 16, 20, 5, 32, 48, 18, 1, 64, 112, 56, 7, 128, 256, 160, 32, 1, 256, 576, 432, 120, 9, 512, 1280, 1120, 400, 50, 1, 1024, 2816, 2816, 1232, 220, 11, 2048, 6144, 6912, 3584, 840, 72, 1, 4096, 13312, 16640, 9984, 2912, 364, 13

Offset: 1

Clark Kimberling, Feb 18 2012

Another version in A201701. - Philippe Deléham, Mar 03 2012
Subtriangle of the triangle given by (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012
Columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976. - Philippe Deléham, Mar 03 2012
Diagonal sums: A052980. - Philippe Deléham, Mar 03 2012

			First seven rows:
   1;
   2,   1;
   4,   3;
   8,   8,  1;
  16,  20,  5,
  32,  48, 18, 1;
  64, 112, 56, 7;
From _Philippe Deléham_, Mar 03 2012: (Start)
Triangle A201701 begins:
   1;
   1,   0;
   2,   1,  0;
   4,   3,  0, 0;
   8,   8,  1, 0, 0;
  16,  20,  5, 0, 0, 0;
  32,  48, 18, 1, 0, 0, 0;
  64, 112, 56, 7, 0, 0, 0, 0;
  ... (End)

Cf. A028297, A207538, A133156.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
(* Prepending 1 and with offset 0: *)
Tpoly[n_] := HypergeometricPFQ[{-n/2, -n/2 + 1/2}, {1/2}, x + 1];
Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* Peter Luschny, Feb 03 2021 *)

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x), where u(1,x)=1, v(1,x)=1.  Also, A207537 = |A028297|.
T(n,k) = 2*T(n-1,k) + T(n-2,k-1). - Philippe Deléham, Mar 03 2012
G.f.: -(1+x*y)*x*y/(-1+2*x+x^2*y). - R. J. Mathar, Aug 11 2015
T(n, k) = [x^k] hypergeom([-n/2, -n/2 + 1/2], [1/2], x + 1) provided offset is set to 0 and 1 prepended. - Peter Luschny, Feb 03 2021

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

A081265 Triangle of coefficients of the polynomials a(n, x) = 2a(n-1, x)+ x^2a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 4, 0, 3, 0, 8, 0, 8, 0, 1, 16, 0, 20, 0, 5, 0, 32, 0, 48, 0, 18, 0, 1, 64, 0, 112, 0, 56, 0, 7, 0, 128, 0, 256, 0, 160, 0, 32, 0, 1, 256, 0, 576, 0, 432, 0, 120, 0, 9, 0, 512, 0, 1280, 0, 1120, 0, 400, 0, 50, 0, 1, 1024, 0, 2816, 0, 2816, 0, 1232, 0, 220

Offset: 0

Paul Barry, Mar 15 2003

Unsigned Chebyshev numbers of the first kind. Columns include A011782, A001792, A001793, A001794, A006974.

For the Riordan coefficient triangle for Chebyshev's T-polynomials (decreasing odd or even powers of x) see A039991. - Wolfdieter Lang, Aug 06 2014

			Triangle rows are {1}, {1,0}, {2,0,1}, {4,0,3,0}, {8,0,8,0,1},.... [Corrected by _Philippe Deléham_, Dec 27 2007]
See the unsigned example under A039991. - _Wolfdieter Lang_, Aug 06 2014

Cf. A008310, A039991 (signed).

T(n,k) = [x^k] a(n,x), k = 0, 1, ..., n, with polynomial a(n,x) defined by the recurrence given as name. Its Binet-de Moivre form is a(n, x) = ((1+sqrt(x^2+1))^n + (1-sqrt(x^2+1))^n)/2.

O.g.f. for row polynomials a(n,x): (1-z)/(1 - 2*z - (x*z)^2). Compare with A039991.

Edited. Name and formula clarified. G.f. of row polynomial, and crossref. A039991 added. - Wolfdieter Lang, Aug 06 2014

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

A304635 Triangle T(n,j) read by rows: the number of j-faces in the hypersimplicial decomposition of the unit cube of n dimensions.

Original entry on oeis.org

1, 5, 2, 18, 14, 3, 56, 64, 27, 4, 160, 240, 150, 44, 5, 432, 800, 660, 288, 65, 6, 1120, 2464, 2520, 1456, 490, 90, 7, 2816, 7168, 8736, 6272, 2800, 768, 119, 8, 6912, 19968, 28224, 24192, 13440, 4896, 1134, 152, 9, 16640, 53760, 86400, 86016, 57120, 25920, 7980, 1600, 189, 10

Offset: 1

R. J. Mathar, May 15 2018

			The triangle starts in row n>= for 1<=j<=n as:
  1,
5,2,
18,14,3,
56,64,27,4,
160,240,150,44,5,
432,800,660,288,65,6,
1120,2464,2520,1456,490,90,7,
2816,7168,8736,6272,2800,768,119,8,
6912,19968,28224,24192,13440,4896,1134,152,9,
16640,53760,86400,86016,57120,25920,7980,1600,189,10,

T. Hibi, N. Li, H. Ohsugi, The Face Vector of a Half-Open Hypersimplex, J. Int. Seq. 18 (2015) 15.6.6

Cf. A001793 (column j=1), A001794 (half of column j=2), A006974 (3rd of column j=3), A014106 (subdiagonal).

A304635 := proc(n,j)
        j*2^(n-j-1)*(n+j+2)/(n+1)*binomial(n+1,j+1) ;
end proc:

T(n,j) = j*2^(n-j-1)*(n+j+2)*binomial(n+,j+1)/(n+1).

A375550 Triangle read by rows: T(m, n, k) = binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1) for m = 4.

Original entry on oeis.org

1, 6, 1, 25, 7, 1, 88, 32, 8, 1, 280, 120, 40, 9, 1, 832, 400, 160, 49, 10, 1, 2352, 1232, 560, 209, 59, 11, 1, 6400, 3584, 1792, 769, 268, 70, 12, 1, 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1, 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1

Offset: 0

Peter Luschny, Sep 23 2024

Triangle T(m,n,k) is a Riordan array of the form ((1-x)^(m-1)*(1-2x)^(-m-1), x/(1-x)), for m = 3. - Igor Victorovich Statsenko, Feb 08 2025

			Triangle starts:
  [0]     1;
  [1]     6,     1;
  [2]    25,     7,     1;
  [3]    88,    32,     8,    1;
  [4]   280,   120,    40,    9,    1;
  [5]   832,   400,   160,   49,   10,    1;
  [6]  2352,  1232,   560,  209,   59,   11,   1;
  [7]  6400,  3584,  1792,  769,  268,   70,  12,  1;
  [8] 16896,  9984,  5376, 2561, 1037,  338,  82, 13,  1;
  [9] 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1;
  ...
Seen as an array of the columns:
  [0] 1,  6, 25,  88,  280,  832,  2352,  6400,  16896, ...
  [1] 1,  7, 32, 120,  400, 1232,  3584,  9984,  26880, ...
  [2] 1,  8, 40, 160,  560, 1792,  5376, 15360,  42240, ...
  [3] 1,  9, 49, 209,  769, 2561,  7937, 23297,  65537, ...
  [4] 1, 10, 59, 268, 1037, 3598, 11535, 34832, 100369, ...
  [5] 1, 11, 70, 338, 1375, 4973, 16508, 51340, 151709, ...
  [6] 1, 12, 82, 420, 1795, 6768, 23276, 74616, 226325, ...

Column k: A055585 (k=0), A001794 (k=1), A001789 (k=2), A027608 (k=3), A055586 (k=4).

Cf. A145018 (diagonal n-2), A375549 (row sums), A049612 (alternating row sums), A122433.

T := (m, n, k) -> binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1);
for n from 0 to 9 do seq(simplify(T(4, n, k)), k = 0..n) od;
# As a binomial sum:
T := (m, n, k) -> add(binomial(m + j, m)*binomial(n + 1, n - (j + k)), j = 0..n-k):
for n from 0 to 9 do [n], seq(T(3, n, k), k = 0..n) od;
# Alternative, generating the array of the columns:
cgf := k -> (1 - x)^(2 - k) / (1 - 2*x)^4:
ser := (k, len) -> series(cgf(k), x, len + 2):
Tcol := (k, len) -> seq(coeff(ser(k, len), x, j), j = 0..len):
seq(lprint([k], Tcol(k, 8)), k = 0..6);

T(m, n, k) = Sum_{j=0..n-k} binomial(m + j, m)*binomial(n + 1, n - (j + k)) for m = 3.

G.f. of column k: (1 - x)^(2 - k) / (1 - 2*x)^4.

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A055584 Triangle of partial row sums (prs) of triangle A055252.

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A207537 Triangle of coefficients of polynomials u(n,x) jointly generated with A207538; see Formula section.

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

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A081265 Triangle of coefficients of the polynomials a(n, x) = 2*a(n-1, x)+ x^2*a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

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

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A304635 Triangle T(n,j) read by rows: the number of j-faces in the hypersimplicial decomposition of the unit cube of n dimensions.

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A375550 Triangle read by rows: T(m, n, k) = binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1) for m = 4.

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A081265 Triangle of coefficients of the polynomials a(n, x) = 2a(n-1, x)+ x^2a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.