A097801 -id:A097801 - cp's OEIS frontend

A341356 The most significant digit in A097801-base.

0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Antti Karttunen, Feb 23 2021

A097801-base uses values 1, 2, 2*3, 2*3*5, 2*3*5*7, 2*3*5*7*9, 2*3*5*7*9*11, 2*3*5*7*9*11*13, 2*3*5*7*9*11*13*15, ..., for its digit-positions, instead of primorials (A002110), thus up to 1889 = 2*3*5*7*9 - 1 = 9*A002110(4) - 1 its representation is identical with the primorial base A049345. Therefore this sequence differs from A276153 for the first time at n=1890, where a(1890)=1, while A276153(1890)=9, as 1890 = 9*A002110(4).

Therefore this sequence might be produced as a rough approximation of A276153 by naive machine learning/mining algorithms. - Antti Karttunen, Mar 09 2021

			In A097801-base, where the digit-positions are given by 1 and the terms of A097801 from its term a(1) onward: 2, 6, 30, 210, 1890, 20790, 270270, 4054050, ..., number 29 is expressed as "421" as 29 = 4*6 + 2*2 + 1*1, thus a(29) = 4. In the same base, number 30 is expressed as "1000" as 30 = 1*30, thus a(30) = 1.
Number 1890 = 2*3*5*7*9 is expressed as "100000", thus a(1890) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A097801.
Cf. A341513 (sum of digits in the same base), A341514 (number of trailing zeros).
Cf. also A002110, A049345.
Differs from similarly constructed A276153 for the first time at n=1890, where a(1890)=1, while A276153(1890)=9.
Differs from similarly constructed A099564 for the first time at n=210, where a(210)=1, while A099564(210)=7.

Block[{nn = 105, b}, b = MixedRadix@ NestWhile[Prepend[#1, 2 #2 - 1] & @@ {#, Length[#] + 1} &, {2}, Times @@ # < nn &]; Array[First@ IntegerDigits[#, b] &, nn + 1, 0]] (* Michael De Vlieger, Feb 23 2021 *)

A341356(n) = { my(m=2, k=3); while(n>=m, n \= m; m = k; k += 2); (n); }; \\ Antti Karttunen & Kevin Ryde, Feb 24 2021

A341513 Sum of digits in A097801-base.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7

Offset: 0

Antti Karttunen, Feb 23 2021

A097801-base uses values 1, 2, 2*3, 2*3*5, 2*3*5*7, 2*3*5*7*9, 2*3*5*7*9*11, 2*3*5*7*9*11*13, 2*3*5*7*9*11*13*15, ..., for its digit-positions, instead of primorials (A002110), thus up to 1889 = 2*3*5*7*9 - 1 = 9*A002110(4) - 1 its representation is identical with the primorial base A049345. Thus this sequence differs from A276150 for the first time at n=1890, where a(1890)=1, while A276150(1890)=9, as 1890 = 9*A002110(4).

			In A097801-base, where the digit-positions are given by 1 and the terms of A097801 from its term a(1) onward: 2, 6, 30, 210, 1890, 20790, 270270, 4054050, ..., number 29 is expressed as "421" as 29 = 4*6 + 2*2 + 1*1, thus a(29) = 4+2+1 = 7. In the same base, number 30 is expressed as "1000" as 30 = 1*30, thus a(30) = 1, and number 1890 = 2*3*5*7*9 is expressed as "100000", thus a(1890) = 1 also.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A097801, A341356 (most significant digit in the same base).

Cf. also A002110, A049345, A276150.

Block[{nn = 105, b}, b = MixedRadix@ NestWhile[Prepend[#1, 2 #2 - 1] & @@ {#, Length[#] + 1} &, {2}, Times @@ # < nn &]; Array[Total@ IntegerDigits[#, b] &, nn + 1, 0]] (* Michael De Vlieger, Feb 23 2021 *)

A341513(n) = { my(u=0,m=2,k=3); while(n, u += n%m; n \= m; m = k; k += 2); (u); };

A341514 Number of trailing zeros in A097801-base.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0

Offset: 1

Antti Karttunen, Feb 25 2021

A097801-base uses values 1, 2, 2*3, 2*3*5, 2*3*5*7, 2*3*5*7*9, 2*3*5*7*9*11, 2*3*5*7*9*11*13, 2*3*5*7*9*11*13*15, ..., for its digit-positions, instead of primorials (A002110), thus up to 1889 = 2*3*5*7*9 - 1 = 9*A002110(4) - 1 its representation is identical with the primorial base A049345.
From Amiram Eldar, Mar 10 2021: (Start)
The asymptotic density of the occurrences of k is 1/2 if k=0, and 2*k/(A097801(k+1)) otherwise.
The asymptotic mean of this sequence is sqrt(e*Pi/2)*erf(1/sqrt(2))/2 = 0.7053430673..., where erf(x) is the error function. (End)

			In A097801-base number 1890 = 2*3*5*7*9 is expressed as "100000", thus a(1890) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A097801, A341356, A341513.

Differs from A276084 for the first time at n=1890, as a(1890) = 5, while A276084(1890) = 4.

Block[{nn = 105, b}, b = MixedRadix@ NestWhile[Prepend[#1, 2 #2 - 1] & @@ {#, Length[#] + 1} &, {2}, Times @@ # < nn &]; Array[LengthWhile[Reverse@ IntegerDigits[#, b], # == 0 &] &, nn]] (* Michael De Vlieger, Feb 25 2021 *)

A341514(n) = { my(m=2,k=3,i=0); while(!(n%m), n /= m; m = k; k += 2; i++); (i); };

For odd n, a(n) = 0; for even n, a(n) = the largest k such that A097801(k) divides n.

A167576 The first column of the ED3 array A167572.

Original entry on oeis.org

1, 5, 23, 167, 1473, 16413, 211479, 3192975, 54010305, 1030249845, 21566327895, 497334999735, 12405876372225, 335591130336525, 9716331072597975, 301633179343890975, 9941514351641143425, 348336799875365041125

Offset: 1

Johannes W. Meijer, Nov 10 2009

Basically a(n) measures the difference between the Euler factorial n! and the Luschny factorial L(n) at half-integer values. For the Luschny factorial see the link. The formula given in the Maple section is a variant of a formula given by Cyril Damamme in A135457. - Peter Luschny, Jul 18 2015

			G.f. = x + 5*x^2 + 23*x^3 + 167*x^4 + 1473*x^5 + 16413*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 1..400
Peter Luschny, Is the Gamma function misdefined? Hadamard versus Euler - Who found the better Gamma function?

Equals the first column of the ED3 array A167572.
Equals the first right hand column of A167583.
Other columns are A167577 and A167578.
Cf. A097801 (the 2*(-1)^n*(2*n-5)!! factor).
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor).
Cf. A024199 and A135457.

L := x -> (1+x*(Psi(1-x/2)-Psi(1/2-x/2)))/(-x)!:
a := x -> (L(x-1/2)-(x-1/2)!)*2^(x-1)*sqrt(Pi):
seq(simplify(a(n)),n=1..18); # Peter Luschny, Jul 18 2015
a := proc(n) option remember: if n=1 then 1 else (2*n-1)*a(n-1)+2*(-1)^n*doublefactorial(2*n-5) fi: end: seq(a(n),n=1..18); # Johannes W. Meijer, Jul 20 2015

a[ n_] := If[ n < 1, 0, (2 n - 3)!! ((-1)^n - I (4 n - 2) Sum[ I^k / k, {k, 1, 2 n - 1, 2}])]; (* Michael Somos, Jul 20 2015 *)
a[ n_] := If[ n < 1, 0, (2 n - 3)!! ((-1)^n + (4 n - 2) Sum[ KroneckerSymbol[ -4, k]/ k, {k, 2 n - 1}])]; (* Michael Somos, Jan 31 2019 *)

{a(n) = if( n<1, 0, prod(k=1, n-1, 2*k - 1) * ((-1)^n - (4*n - 2) * sum(k=1, n, (-1)^k / (2*k - 1))))}; /* Michael Somos, Jul 20 2015 */

a(n) = (-1)^n*(2*n-3)!!*(1 + (4*n-2)*Sum_{k=0..n-1} (-1)^(k+n)/(2*k+1)).
a(n) = (2*n-1)*a(n-1) + 2*(-1)^n*(2*n-5)!! with a(1) = 1.
a(n) = 4*a(n-1) + (4*n^2 - 16*n + 15)*a(n-2) with a(1) = 1 and a(2) = 5 [Superseeker].
0 = a(n)*a(n+1)*(-440*a(n+2) - 220*a(n+3) + 55*a(n+4)) + a(n)*a(n+2)*(536*a(n+2) - 118*a(n+3) - 4*a(n+4)) + a(n)*a(n+3)*(-4*a(n+3) + a(n+4)) + a(n+1)^2*(-220*a(n+2) - 32*a(n+3) + 8*a(n+4)) + a(n+1)*a(n+2)*(+71*a(n+2) + 4*a(n+3) - 2*a(n+4)) + a(n+2)^2*(-4*a(n+2) + a(n+3)) if n>0. - Michael Somos, Jul 19 2015
a(n) = (-1 + (n-1/2)*LerchPhi(-1,1,n+1/2) + (-n+1/2)*LerchPhi(-1,1,-n+1/2))/(1-2*n)!!. - Johannes W. Meijer, Jul 20 2015
a(n) = A024199(n) + A135457(n). - Cyril Damamme, Jul 22 2015
a(n) = ((-1)^n/(2*n - 1) + Pi/2 - (-1)^n LerchPhi(-1, 1, n + 1/2)) (2*n - 1)!!. - Michael Somos, Jan 31 2019

A068215 Denominator of Borwein integral of order 2n+1, as defined by Weisstein.

Original entry on oeis.org

2, 6, 30, 210, 1890, 20790, 270270, 1896516717212415135141110350293750000, 1220462921565155916674902677397230198502690752000000000

Offset: 0

Eric W. Weisstein, Feb 21 2002

Eric Weisstein's World of Mathematics, Borwein Integrals
Wikipedia, Borwein integral [From _N. J. A. Sloane_, Feb 25 2012]

Cf. A068214 (supposed numerators), A144616 (denominators of the conventional Borwein integrals).

i[n_] := Times@@(Sin[x/# ]&/@Range[1, n, 2])/x^((n+1)/2)/Pi; Denominator[Table[Integrate[i[n], {x, 0, \[Infinity]}], {n, 1, 19, 2}]]

a(n) = A144616(n+1)*A097801(n+1) [assuming that the numerators are really A068214]. - Andrey Zabolotskiy, Oct 18 2016

Name edited by Andrey Zabolotskiy, Dec 14 2024

A129779 a(1) = 1, a(2) = -1, a(3) = 2; for n > 3, a(n) = -(2n-5)a(n-1).

Original entry on oeis.org

1, -1, 2, -6, 30, -210, 1890, -20790, 270270, -4054050, 68918850, -1309458150, 27498621150, -632468286450, 15811707161250, -426916093353750, 12380566707258750, -383797567925021250, 12665319741525701250

Offset: 1

Paul Curtz, May 17 2007

sign

Sequence is also the first column of the inverse of the infinite lower triangular matrix M, where M(j,k) = 1+2*(k-1)*(j-k) for k < j, M(j,k) = 1 for k = j, M(j,k) = 0 for k > j.
Upper left 6 X 6 submatrix of M is
  [ 1  0  0  0  0  0 ]
  [ 1  1  0  0  0  0 ]
  [ 1  3  1  0  0  0 ]
  [ 1  5  5  1  0  0 ]
  [ 1  7  9  7  1  0 ]
  [ 1  9 13 13  9  1 ],
and upper left 6 X 6 submatrix of M^-1 is
  [    1    0    0    0    0    0 ]
  [   -1    1    0    0    0    0 ]
  [    2   -3    1    0    0    0 ]
  [   -6   10   -5    1    0    0 ]
  [   30  -50   26   -7    1    0 ]
  [ -210  350 -182   50   -9    1 ].
Row sums of M are 1, 2, 5, 12, 25, 46, ... (see A116731); diagonal sums of M are 1, 1, 2, 4, 7, 13, 20, 32, 45, 65, 86, 116, 147, 189, ... with first differences 0, 1, 2, 3, 6, 7, 12, 13, 20, 21, 30, 31, 42, ... and second differences 1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, ... (see A093178).

G. C. Greubel, Table of n, a(n) for n = 1..400

Cf. A093178, A097801, A116731.

F:=Factorial;; Concatenation([1,-1], List([3..25], n-> (-1)^(n+1)*F(2*n-5)/(2^(n-4)*F(n-3)) )); # G. C. Greubel, Nov 25 2019

m:=19; M:=Matrix(IntegerRing(), m, m, [< j, k, Maximum(0, 1+2*(k-1)*(j-k)) > : j, k in [1..m] ] ); Transpose(ColumnSubmatrix(M^-1, 1, 1)); // Klaus Brockhaus, May 21 2007

F:=Factorial; [1,-1] cat [(-1)^(n+1)*F(2*n-5)/(2^(n-4)*F(n-3)): n in [3..25]]; // G. C. Greubel, Nov 25 2019

seq(`if`(n<3, (-1)^(n-1), (-1)^(n-1)*(2*n-5)!/(2^(n-4)*(n-3)!)), n=1..25); # G. C. Greubel, Nov 25 2019

a[n_]:= -(2*n-5)*a[n-1]; a[1]=1; a[2]=-1; a[3]=2; Array[a, 20] (* Robert G. Wilson v *)
Table[If[n<3, (-1)^(n-1), (-1)^(n+1)*(2*n-5)!/(2^(n-4)*(n-3)!)], {n,25}] (* G. C. Greubel, Nov 25 2019 *)

{m=19; print1(1, ",", -1, ","); print1(a=2, ","); for(n=4, m, k=-(2*n-5)*a; print1(k, ","); a=k)} \\ Klaus Brockhaus, May 21 2007

{print1(1, ",", -1, ","); for(n=3, 19, print1((-1)^(n-1)*(2*(n-2))!/((n-2)!*2^(n-3)), ","))} \\ Klaus Brockhaus, May 21 2007

{m=19; M=matrix(m, m, j, k, if(k>j, 0, if(k==j, 1, 1+2*(k-1)*(j-k)))); print((M^-1)[, 1]~)} \\ Klaus Brockhaus, May 21 2007

f=factorial; [1,-1]+[(-1)^(n+1)*f(2*n-5)/(2^(n-4)*f(n-3)) for n in (3..25)] # G. C. Greubel, Nov 25 2019

a(n) = (-1)^(n-1)*A097801(n-2) = (-1)^(n-1)*(2*(n-2))!/((n-2)!*2^(n-3)) for n > 2, with a(1)=1, a(2)=-1.

G.f.: 1 + x - x*W(0) , where W(k) = 1 + 1/( 1 - x*(2*k+1)/( x*(2*k+1) - 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 22 2013

Edited and extended by Klaus Brockhaus and Robert G. Wilson v, May 21 2007

A097749 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows. Let A(n,k) be the triangle in A097474. Then T(n,k) is defined by the orthogonality relations Sum_{j=i..r} T(r,j)A(j,i)2^-floor((j+3)/2) = 0 if i != r, = (2r+1)!/(r!*2^r) if i = r.

Original entry on oeis.org

2, 1, 2, -1, 10, 6, 5, -35, 105, 30, -63, 420, -882, 1260, 210, 1576, -10395, 20790, -20790, 17325, 1890, -68409, 450450, -891891, 849420, -495495, 270270, 20790, 4729726, -31126095, 61486425, -57972915, 32207175, -12297285, 4729725, 270270

Offset: 0

N. J. A. Sloane, Sep 21 2004

			Triangle begins:
2
1 2
-1 10 6
5 -35 105 30
-63 420 -882 1260 210

H. W. Gould, Power sum identities for arbitrary symmetric arrays, SIAM J. Appl. Math., 17 (1969), 307-316.

Cf. A097474, A097801. Row sums give A001147. Is the left-hand edge A004193?

More terms from Sean A. Irvine, Mar 25 2013

A126063 Triangle read by rows: see A128196 for definition.

Original entry on oeis.org

1, 1, 2, 3, 6, 4, 15, 30, 20, 8, 105, 210, 140, 56, 16, 945, 1890, 1260, 504, 144, 32, 10395, 20790, 13860, 5544, 1584, 352, 64, 135135, 270270, 180180, 72072, 20592, 4576, 832, 128, 2027025, 4054050, 2702700, 1081080, 308880, 68640, 12480, 1920, 256

Offset: 0

N. J. A. Sloane, Feb 28 2007

			Triangle begins:
       1
       1,       2
       3,       6,       4
      15,      30,      20,       8
     105,     210,     140,      56,     16
     945,    1890,    1260,     504,    144,    32
   10395,   20790,   13860,    5544,   1584,   352,    64
  135135,  270270,  180180,   72072,  20592,  4576,   832,  128

Ivan Neretin, Rows n = 0..100, flattened
P. Luschny, Variants of Variations.

First column is A001147, second column is A097801.

The diagonal is A000079, the subdiagonal is A014480.

A126063 := (n,k) -> 2^k*doublefactorial(2*n-1)/ doublefactorial(2*k-1); seq(print(seq(A126063(n,k),k=0..n)),n=0..7); # Peter Luschny, Dec 20 2012

Flatten[Table[2^k (2n - 1)!!/(2k - 1)!!, {n, 0, 8}, {k, 0, n}]] (* Ivan Neretin, May 11 2015 *)

Let H be the diagonal matrix diag(1,2,4,8,...) and
let G be the matrix (n!! defined as A001147(n), -1!! = 1):
(-1)!!/(-1)!!
1!!/(-1)!! 1!!/1!!
3!!/(-1)!! 3!!/1!! 3!!/3!!
5!!/(-1)!! 5!!/1!! 5!!/3!! 5!!/5!!
...
Then T = G*H. [Gottfried Helms]
T(n,k) = 2^k*(2n - 1)!!/(2k - 1)!!. - Ivan Neretin, May 13 2015

A193561 Augmentation of the triangle A004736. See Comments.

Original entry on oeis.org

1, 2, 1, 6, 6, 3, 24, 36, 30, 15, 120, 240, 270, 210, 105, 720, 1800, 2520, 2520, 1890, 945, 5040, 15120, 25200, 30240, 28350, 20790, 10395, 40320, 141120, 272160, 378000, 415800, 374220, 270270, 135135, 362880, 1451520, 3175200, 4989600

Offset: 0

Clark Kimberling, Jul 30 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding A193561, if the triangle is written as (w(n,k)), then
w(n,n)=A001147(n), "double factorial numbers";
w(n,n-1)=A097801(n), (2n)!/(n!*2^(n-1))
col 1:  A000142, n!
col 2: A001286, Lah numbers, (n-1)*n!/2

			First 5 rows of A193560:
1
2.....1
6.....6....3
24....36...30...15
120...240..270..210..105

Cf. A193091.

p[n_, k_] := n + 1 - k
Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* A004736 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]]  (* A193561 *)
Flatten[Table[v[n], {n, 0, 8}]]

A238087 A129779(n+2)/A006954(n).

Original entry on oeis.org

-1, 1, -1, 1, -5, 63, -315, 99, -675675, 135135, -1640925, 83329155, -4583103525, 5791834125, -71152682225625, 14230536445125, -26797763435625, 24833960277501375, -73881031825566590625, 8546099690638125, -106610328924292590271875

Offset: 0

Paul Curtz, Feb 24 2014

sign

In A129779 or A097801 there is no mention of a link with the Bernoulli numbers.

It appears that a(n+6) is divisible by 4*(n+4)^2-1 and that the sum of the digits of a(n+5) is a multiple of 9.

			a(0) = -1/1, a(1) = 2/2, a(2) = -6/6, a(3) = 30/30, a(4) = -210/42 =-5.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000466.

a[0] = -1; a[1] = 1; a[n_] := (-1)^(n+1)*(2*n)!/(2^(n-1)*n!*Denominator[BernoulliB[2*n-2]]); Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2014 *)

a(n+2) = -A004193(n+1)/A000367(n+1).

More terms from Jean-François Alcover, Feb 24 2014

cp's OEIS Frontend

A341356 The most significant digit in A097801-base.

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A341513 Sum of digits in A097801-base.

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A341514 Number of trailing zeros in A097801-base.

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A167576 The first column of the ED3 array A167572.

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A068215 Denominator of Borwein integral of order 2n+1, as defined by Weisstein.

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A129779 a(1) = 1, a(2) = -1, a(3) = 2; for n > 3, a(n) = -(2*n-5)*a(n-1).

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A097749 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows. Let A(n,k) be the triangle in A097474. Then T(n,k) is defined by the orthogonality relations Sum_{j=i..r} T(r,j)*A(j,i)*2^-floor((j+3)/2) = 0 if i != r, = (2r+1)!/(r!*2^r) if i = r.

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A129779 a(1) = 1, a(2) = -1, a(3) = 2; for n > 3, a(n) = -(2n-5)a(n-1).

A097749 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows. Let A(n,k) be the triangle in A097474. Then T(n,k) is defined by the orthogonality relations Sum_{j=i..r} T(r,j)A(j,i)2^-floor((j+3)/2) = 0 if i != r, = (2r+1)!/(r!*2^r) if i = r.