A091885 -id:A091885 - cp's OEIS frontend

A385343 Exponential Riordan array (1, arcsin(x)).

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

Offset: 0

Seiichi Manyama, Jun 26 2025

			Triangle starts:
  1;
  0,   1;
  0,   0,  1;
  0,   1,  0,   1;
  0,   0,  4,   0,  1;
  0,   9,  0,  10,  0,  1;
  0,   0, 64,   0, 20,  0, 1;
  0, 225,  0, 259,  0, 35, 0, 1;

Essentialy same as A121408.
Row sums give A006228.
Cf. A091885, A136630, A185951.

T(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(asin(x)^k/k!, n);

E.g.f. of column k (with leading zeros): arcsin(x)^k / k!

T(n,k) = A121408(n,k) for k > 0.

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

A160563 Table of the number of (n,k)-Riordan complexes, read by rows.

Original entry on oeis.org

1, 1, 1, 9, 10, 1, 225, 259, 35, 1, 11025, 12916, 1974, 84, 1, 893025, 1057221, 172810, 8778, 165, 1, 108056025, 128816766, 21967231, 1234948, 28743, 286, 1, 18261468225, 21878089479, 3841278805, 230673443, 6092515, 77077, 455, 1, 4108830350625, 4940831601000

Offset: 0

Jonathan Vos Post, May 19 2009

From Table 4, right-hand side, of Gelineau and Zeng.

Essentially a row-reversal of A008956. - R. J. Mathar, May 20 2009

			Triangle starts:
  [0]         1;
  [1]         1,          1;
  [2]         9,         10,        1;
  [3]       225,        259,       35,        1;
  [4]     11025,      12916,     1974,       84,     1;
  [5]    893025,    1057221,   172810,     8778,   165,    1;
  [6] 108056025,  128816766, 21967231,  1234948, 28743,  286, 1;
.
For row 3: F(x) := 1/cos(x). Then 225*F(x) + 259*(d/dx)^2(F(x)) + 35*(d/dx)^4(F(x)) + (d/dx)^6(F(x)) = 720*(1/cos(x))^7, where F^(r) denotes the r-th derivative of F(x).

Yoann Gelineau and Jiang Zeng, Combinatorial Interpretations of the Jacobi-Stirling Numbers, arXiv:0905.2899 [math.CO], May 2009.
W. Zhang, Some identities involving the Euler and the central factorial numbers, The Fibonacci Quarterly, Vol. 36, Number 2, May 1998.

Cf. A001819, A008275, A008277, A160562, A091885, A182867.

t := proc(n,k) option remember ; expand(x*mul(x+n/2-i,i=1..n-1)) ; coeftayl(%,x=0,k) ; end:
v := proc(n,k) option remember ; 4^(n-k)*t(2*n+1,2*k+1) ; end:
A160563 := proc(n,k) abs(v(n,k)) ; end: for n from 0 to 10 do for k from 0 to n do printf("%d,",A160563(n,k)) ; od: od: # R. J. Mathar, May 20 2009
# Using a bivariate generating function (albeit generating signed terms):
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, k):
OddPart := (T, len) -> local n, k;
seq(print(seq(T(n, k), k = 1..n, 2)), n = 1..2*len, 2):
OddPart(T, 6);  # Peter Luschny, Mar 03 2024

t[, 0] = 1; t[n, n_] := t[n, n] = ((2*n - 1)!!)^2; t[n_, k_] := t[n, k] = (2*n - 1)^2*t[n - 1, k - 1] + t[n - 1, k];
T[n_, k_] := t[n, n - k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2017, after R. J. Mathar's comment *)

a(n,k) = |v(n,k)| where v(n,k) = v(n-1,k-1) - (2n-1)^2*v(n-1,k); eq (4.2).
Let F(x) = 1/cos(x). Then (2*n)!*(1/cos(x))^(2*n+1) = Sum_{k=0..n} T(n,k)*F^(2*k)(x), where F^(r) denotes the r-th derivative of F(x) (Zhang 1998). An example is given below. - Peter Bala, Feb 06 2012
Given a (0, 0)-based triangle U we call the triangle [U(n, k), k=1..n step 2, n=1..len step 2] the 'odd subtriangle' of U. This triangle is the odd subtriangle of U(n, k) = n! * [x^(n-k)] [t^n] (t + sqrt(1 + t^2))^x, albeit with signed terms. See A182867 for the even subtriangle. - Peter Luschny, Mar 03 2024

Extended by R. J. Mathar, May 20 2009

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A385343 Exponential Riordan array (1, arcsin(x)).

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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!.

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A160563 Table of the number of (n,k)-Riordan complexes, read by rows.

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cp's OEIS Frontend

A385343 Exponential Riordan array (1, arcsin(x)).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

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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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A160563 Table of the number of (n,k)-Riordan complexes, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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!.