A182971 -id:A182971 - cp's OEIS frontend

A181888 Second column of triangle in A182971.

0, 0, 9, 64, 259, 784, 1974, 4368, 8778, 16368, 28743, 48048, 77077, 119392, 179452, 262752, 375972, 527136, 725781, 983136, 1312311, 1728496, 2249170, 2894320, 3686670, 4651920, 5818995, 7220304, 8892009, 10874304, 13211704, 15953344, 19153288, 22870848, 27170913, 32124288, 37808043, 44305872, 51708462, 60113872

Offset: 1

N. J. A. Sloane, Feb 01 2011

nonn

These are scaled central factorial numbers (see A182971).

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

Cf. A182971. Bisections give A001823 and A181889.

CoefficientList[Series[x^3(9+x)/(1-x)^7,{x,0,40}],x] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,0,0,9,64,259,784},50] (* Harvey P. Dale, Jul 03 2013 *)

G.f.: x^3*(9+x)/(1-x)^7.

a(1)=0, a(2)=0, a(3)=0, a(4)=9, a(5)=64, a(6)=259, a(7)=784, a(n) = 7*a(n-1) -21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Jul 03 2013

A184878 Third column (negated) of triangle in A182971.

Original entry on oeis.org

0, 0, 0, 0, 225, 2304, 12916, 52480, 172810, 489280, 1234948, 2846272, 6092515, 12263680, 23434840, 42828032, 75296468, 127959680, 211022360, 338814080, 531091781, 814651904, 1225304300, 1810265600, 2631035550, 3766825920, 5318617980, 7413931200, 10212392775, 13912204800, 18757613424, 25047492096, 33145159080, 43489557760, 56607936880, 73130176768, 93804916777, 119517648640, 151310950180, 190407043840

Offset: 1

N. J. A. Sloane, Feb 01 2011

nonn

These are scaled central factorial numbers.

Cf. A182971.

A184877 a(n) = n^2(n-2)^2(n-4)^2...(1 or 2)^2.

Original entry on oeis.org

1, 1, 4, 9, 64, 225, 2304, 11025, 147456, 893025, 14745600, 108056025, 2123366400, 18261468225, 416179814400, 4108830350625, 106542032486400, 1187451971330625, 34519618525593600, 428670161650355625, 13807847410237440000, 189043541287806830625, 6682998146554920960000, 100004033341249813400625

Offset: 0

N. J. A. Sloane, Feb 01 2011

nonn

			a(0) = Empty product = 1;
a(1) = 1^2 = 1;
a(2) = 2^2 = 4;
a(3) = 3^2*1^2 = 9;
a(4) = 4^2*2^2 = 64;
a(5) = 5^2*3^2*1^2 = 225;
...

David A. Corneth, Table of n, a(n) for n = 0..449

Rightmost diagonal of A182971.
With signs, a row of A288580.
Cf. A006882, A197036, A197037.

[1] cat [(&*[(n-2*k)^2: k in [0..Floor((n-1)/2)]]): n in [1..50]]; // G. C. Greubel, Oct 14 2018

Table[Product[(n-2*k)^2, {k,0,Floor[(n-1)/2]}], {n,0,50}] (* G. C. Greubel, Oct 14 2018 *)

vector(100, n, n--; prod(k=0, (n-1)\2, (n-2*k)^2)) \\ Altug Alkan, Oct 29 2015

first(n) = {if(n<2, return(vector(n, i, 1))); my(res = vector(n), i = 3); res[1] = res[2] = 1; while(i<=n, res[i] = res[i-2]*(i-1)^2; i++) ;res} \\ David A. Corneth, Aug 03 2017

a(n) = (n!!)^2 = A006882(n)^2. - Gionata Neri, Oct 29 2015
For n > 1, a(n) = n^2 * a(n-2). - David A. Corneth, Aug 03 2017
From Amiram Eldar, Apr 09 2022: (Start)
Sum_{n>=0} 1/a(n) = BesselI(0, 1) + StruveL(0, 1)*Pi/2 = A197036 + A197037 * Pi/2.
Sum_{n>=0} (-1)^n/a(n) = BesselI(0, 1) - StruveL(0, 1)*Pi/2. (End)
E.g.f.: 1/(1-x^2) + x*(1+arcsin(x))/(1-x^2)^(3/2). - Fabián Pereyra, May 14 2023

A121408 Triangle T(n,k) defined by the generating function: exp(yarcsin(x))-1 = Sum_{n>=1} (Sum_{k=1..n} T(n,k)y^k)*x^n/n!.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 9, 0, 10, 0, 1, 0, 64, 0, 20, 0, 1, 225, 0, 259, 0, 35, 0, 1, 0, 2304, 0, 784, 0, 56, 0, 1, 11025, 0, 12916, 0, 1974, 0, 84, 0, 1, 0, 147456, 0, 52480, 0, 4368, 0, 120, 0, 1, 893025, 0, 1057221, 0, 172810, 0, 8778, 0, 165, 0, 1, 0, 14745600, 0

Offset: 1

Emeric Deutsch, Jul 28 2006

Row sums are equal to A006228(n). This is sequence A091885 with additional intertwining zeros.
F(n,m) = n!*T(n,m)/m! is a composite (akin to Riordan arrays) of F(x)=arcsin(x) and (F(x))^m = Sum_{n>=m} F(n,m)*x^n, and for o.g.f. G(x), G(arcsin(x)) = g(0) +Sum_{n>=1} Sum_{m=1..n} F(n,m)*g(m)*x^n, see the preprint. - Vladimir Kruchinin, Feb 10 2011
The unsigned matrix inverse is A136630 (with a different offset). - Peter Bala, Feb 23 2011
Also the Bell transform of A177145. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle starts:
  1;
  0,1;
  1,0,1;
  0,4,0,1;
  9,0,10,0,1;
  0,64,0,20,0,1;
Row polynomials R(6,x) = x^2*(x^2 + 2^2)*(x^2 + 4^2) = 64*x^2 + 20*x^4 + x^6 and
R(7,x) = x*(x^2 + 1)*(x^2 + 3^2)*(x^2 + 5^2) = 225*x + 259*x^3 + 35*x^5 + x^7. - _Peter Bala_, Aug 29 2012

B. C. Berndt, Ramanujan's Notebooks Part 1, Springer-Verlag 1985.

Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A000111, A000142, A006228, A091885, A136630. A182971.

g:=exp(y*arcsin(x))-1: gser:=simplify(series(g,x=0,15)): for n from 1 to 12 do P[n]:=sort(n!*coeff(gser,x,n)) od: for n from 1 to 12 do seq(coeff(P[n],y,k),k=1..n) od; # yields sequence in triangular form
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n::odd,0,doublefactorial(n-1)^2), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[OddQ[#], 0, (# - 1)!!^2] &, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

T(n,m) = ((n-1)!/(m-1)!) *sum_{k=1..n-m} sum_{j=1..k} binomial(k,j) *(2^(1-j) /(n-m+j)!) *sum{i=0..floor(j/2)} (-1)^((n-m)/2-i-j) *binomial(j,i) *(j-2*i)^(n-m+j) *binomial(k+n-1,n-1), n>m and even(n-m). [Vladimir Kruchinin, Feb 10 2011]
From Peter Bala, Aug 29 2012: (Start)
See A182971 for a version of the row reverse of this triangle.
Even-indexed row polynomial R(2*n,x) = x^2*prod(k=1..n-1, (x^2 + (2*k)^2) ).
Odd-indexed row polynomial R(2*n+1,x) = x*prod(k=1..n, (x^2 + (2*k-1)^2) ). See Berndt p.263. (End)
Sum_{k=0..n} T(n+1,k+1)*A000111(k) = n! = A000142(n). - Alexander Burstein, Aug 01 2025

A182867 Triangle read by rows: row n gives coefficients in expansion of Product_{i=1..n} (x - (2i)^2), highest powers first.

Original entry on oeis.org

1, 1, -4, 1, -20, 64, 1, -56, 784, -2304, 1, -120, 4368, -52480, 147456, 1, -220, 16368, -489280, 5395456, -14745600, 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400, 1, -560, 119392, -12263680, 633721088, -15658639360, 157294854144, -416179814400, 1, -816, 262752, -42828032, 3773223168, -177891237888, 4165906530304, -40683662475264, 106542032486400, 1, -1140, 527136, -127959680, 17649505536, -1400415544320, 61802667606016, -1390437378293760, 13288048674471936, -34519618525593600

Offset: 0

N. J. A. Sloane, Feb 01 2011

These are scaled central factorial numbers (see the discussion in the Comments section of A008955). The coefficients in the expansion of Product_{i=1..n} (x - i^2) give A008955, and the coefficients in the expansion of Product_{i=1..n} (x - (2i+1)^2) give A008956.

			Triangle begins:
 1
 1, -4
 1, -20, 64
 1, -56, 784, -2304
 1, -120, 4368, -52480, 147456
 1, -220, 16368, -489280, 5395456, -14745600
 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400
 1, -560, 119392, -12263680, 633721088, -15658639360, 157294854144, -416179814400
 1, -816, 262752, -42828032, 3773223168, -177891237888, 4165906530304, -40683662475264, 106542032486400
 1, -1140, 527136, -127959680, 17649505536, -1400415544320, 61802667606016, -1390437378293760, 13288048674471936, -34519618525593600
...
For example, for n=2, (x-4)(x-16) = x^2 - 20x + 64 => [1, -20, 64].

T. L. Curtright, D. B. Fairlie, and C. K. Zachos, A compact formula for rotations as spin matrix polynomials, arXiv preprint arXiv:1402.3541 [math-ph], 2014.
T. L. Curtright and T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arXiv:1408.0767 [math-ph], 2014.
T. L. Curtright, More on Rotations as Spin Matrix Polynomials, arXiv preprint arXiv:1506.04648 [math-ph], 2015.

Cf. A008955, A008956. This triangle is formed from the even-indexed rows of A182971 (the odd-indexed rows give A008956).

Cf. A160563.

Q:= n -> if n mod 2 = 0 then sort(expand(mul(x-4*i^2,i=1..n/2)));
else sort(expand(mul(x-(2*i+1)^2,i=0..(n-1)/2))); fi;
for n from 0 to 10 do
t1:=eval(Q(2*n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od:
# Using a bivariate generating function (adding a superdiagonal 1,0,0, ...):
gf := (t + sqrt(1 + t^2))^x:
ser := series(gf, t, 20): ct := n -> coeff(ser, t, n):
T := (n, k) -> n!*coeff(ct(n), x, n - k):
EvenPart := (T, len) -> local n, k;
seq(print(seq(T(n, k), k = 0..n, 2)), n = 0..2*len-1, 2):
EvenPart(T, 6);  # Peter Luschny, Mar 03 2024

Given a (0, 0)-based triangle U we call the triangle [U(n, k), k=0..n step 2, n=0..len step 2] the 'even subtriangle' of U. This triangle is the even subtriangle of U(n, k) = n! * [x^(n-k)] [t^n] (t + sqrt(1 + t^2))^x, albeit adding a superdiagonal 1, 0, 0, ... See A160563 for the odd subtriangle. - Peter Luschny, Mar 03 2024

cp's OEIS Frontend

A181888 Second column of triangle in A182971.

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A184878 Third column (negated) of triangle in A182971.

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A184877 a(n) = n^2*(n-2)^2*(n-4)^2*...*(1 or 2)^2.

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A121408 Triangle T(n,k) defined by the generating function: exp(y*arcsin(x))-1 = Sum_{n>=1} (Sum_{k=1..n} T(n,k)*y^k)*x^n/n!.

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A182867 Triangle read by rows: row n gives coefficients in expansion of Product_{i=1..n} (x - (2i)^2), highest powers first.

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A184877 a(n) = n^2(n-2)^2(n-4)^2...(1 or 2)^2.

A121408 Triangle T(n,k) defined by the generating function: exp(yarcsin(x))-1 = Sum_{n>=1} (Sum_{k=1..n} T(n,k)y^k)*x^n/n!.