A103452 -id:A103452 - cp's OEIS frontend

A103451 Triangular array T read by rows: T(n, 0) = T(n, n) = 1, T(n, k) = 0 for 0 < k < n.

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Feb 06 2005

Equals Pascal's triangle (A007318) where all elements > 1 are replaced with zero. Therefore it might be called "binomial skeleton".

Row sums are in A040000, antidiagonal sums are in A040001. When construed as a lower triangular matrix, the matrix inverse is A103452.

			First few rows are:
  1;
  1, 1;
  1, 0, 1;
  1, 0, 0, 1;
  1, 0, 0, 0, 1;
  1, 0, 0, 0, 0, 1;
  ...

Michael De Vlieger, Rows n = 0..140 of triangle, flattened
Carl M. Bender and Gerald V. Dunne, Polynomials and operator orderings, J. Math. Phys. 29 (1988), 1727-1731.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 18.
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 5.

Cf. A007318, A040000, A040001, A097806, A103452.

r:=14; T:=ScalarMatrix(r, 1); for n in [1..r] do T[n, 1]:=1; end for; &cat[ [ T[n, k]: k in [1..n] ]: n in [1..r] ];

/* As triangle */ [[Binomial(n, k-n)+Binomial(n, -k)-Binomial(0, n+k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 20 2016

Table[Boole[n == 0 || Mod[k, n] == 0], {n, 0, 14}, {k, 0, n}] (* or *)
Table[Binomial[n, k - n] + Binomial[n, -k] - Binomial[0, n + k], {n, 0, 14}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 19 2016 *)

for(n=0,15, for(k=0,n, print1(if(k==0||k==n, 1, 0), ", "))) \\ G. C. Greubel, Dec 08 2018

from math import isqrt, comb
def A103451(n):
    if n==0: return 1
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    return int(not (n-comb(a+1,2))%a) # Chai Wah Wu, Jun 24 2025

def A103451(n,k): return 1 if (k==0 or k==n) else 0
flatten([[A103451(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 14 2021

a(n) = A097806(n-1) for n > 0. - Philippe Deléham, Oct 16 2007
T(n,k) = C(n,k-n) + C(n,-k) - C(0,n+k), 0 <= k <= n. - Eric Werley, Jul 01 2011
From Stefano Spezia, Jul 04 2024: (Start)
G.f.: (1 - x^2*y)/((1 - x)*(1 - x*y)).
E.g.f.: BesselI(0, 2*sqrt(x*y)) + exp(x) - 1. (End)

Edited by Klaus Brockhaus, Jan 26 2011

A103454 a(n) = 0^n + 4^n - 1.

Original entry on oeis.org

1, 3, 15, 63, 255, 1023, 4095, 16383, 65535, 262143, 1048575, 4194303, 16777215, 67108863, 268435455, 1073741823, 4294967295, 17179869183, 68719476735, 274877906943, 1099511627775, 4398046511103, 17592186044415, 70368744177663

Offset: 0

Paul Barry, Feb 06 2005

A transform of 4^n under the matrix A103452.

The square of the cotangent of the arcsin of 1/(2^n). - Al Hakanson (hawkuu(AT)excite.com), Feb 23 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A024036, A103452.

[0^n+4^n-1: n in [0..30]]; // Vincenzo Librandi, Jul 02 2011

Table[Boole[n==0] +4^n -1, {n,0,40}] (* G. C. Greubel, Jun 21 2021 *)

[1]+[4^n -1 for n in [1..40]] # G. C. Greubel, Jun 21 2021

G.f.: (1 - 2*x + 4*x^2)/((1-x)*(1-4*x));
a(n) = Sum_{k=0..n} A103452(n, k)*4^k;
a(n) = Sum_{k=0..n} (2*0^(n-k) - 1)*0^(k*(n-k))4^k.
a(n) = A024036(n), n > 0. - R. J. Mathar, Aug 30 2008
E.g.f.: 1 - exp(x) + exp(4*x). - G. C. Greubel, Jun 21 2021
a(n) = 5*a(n-1) - 4*a(n-2). - Wesley Ivan Hurt, Mar 17 2023

A206735 Triangle T(n,k), read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 4, 1, 0, 5, 10, 10, 5, 1, 0, 6, 15, 20, 15, 6, 1, 0, 7, 21, 35, 35, 21, 7, 1, 0, 8, 28, 56, 70, 56, 28, 8, 1, 0, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Philippe Deléham, Feb 11 2012

A103452*A007318 as infinite lower triangular matrices.

Essentially the same as A199011.

			Triangle begins :
1
0, 1
0, 2, 1
0, 3, 3, 1
0, 4, 6, 4, 1
0, 5, 10, 10, 5, 1
0, 6, 15, 20, 15, 6, 1
0, 7, 21, 35, 35, 21, 7, 1
0, 8, 28, 56, 70, 56, 28, 8, 1
0, 9, 36, 84, 126, 126, 84, 36, 9, 1
0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1
0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Cf. A007318, A000071 (antidiagonal sums).

T(n,k) = A007318(n,k) - A073424(n,k).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (1+x)^n - 1 + 0^n.
T(n,0) = 0^n = A000007(n), T(n,k) = binomial(n,k) for k>0.
G.f.: (1-2*x+(1+y)*x^2)/(1-2x+(1+y)*x^2-y*x).
Sum{k, 0<=k<=n} T(n,k)^x = A000027(n+1), A000225(n), A030662(n), A096191(n), A096192(n) for x = 0, 1, 2, 3, 4 respectively.

A103455 a(n) = 0^n + 5^n - 1.

Original entry on oeis.org

1, 4, 24, 124, 624, 3124, 15624, 78124, 390624, 1953124, 9765624, 48828124, 244140624, 1220703124, 6103515624, 30517578124, 152587890624, 762939453124, 3814697265624, 19073486328124, 95367431640624, 476837158203124

Offset: 0

Paul Barry, Feb 06 2005

A transform of 5^n under the matrix A103452.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A024049, A103452.

[0^n+5^n-1: n in [0..30]]; // Vincenzo Librandi, Jun 06 2011

Join[{1},5^Range[20]-1] (* Harvey P. Dale, Nov 15 2011 *)

[1]+[5^n -1 for n in [1..40]] # G. C. Greubel, Jun 21 2021

G.f.: (1 - 2*x + 5*x^2)/((1-x)*(1-5*x)).
a(n) = Sum_{k=0..n} A103452(n, k)*5^k.
a(n) = Sum_{k=0..n} (2*0^(n-k) - 1)*0^(k*(n-k))*5^k.
a(n) = A024049(n), n > 0. - R. J. Mathar, Aug 30 2008
E.g.f.: 1 - exp(x) + exp(5*x). - G. C. Greubel, Jun 21 2021

A103453 a(n) = 0^n + 3^n - 1.

Original entry on oeis.org

1, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, 59048, 177146, 531440, 1594322, 4782968, 14348906, 43046720, 129140162, 387420488, 1162261466, 3486784400, 10460353202, 31381059608, 94143178826, 282429536480, 847288609442

Offset: 0

Paul Barry, Feb 06 2005

A transform of 3^n under the matrix A103452.

a(n) is the number of moves required to solve a Towers of Hanoi puzzle of 3 towers in a line (no direct connection between the two towers on the ends) with n pieces to be moved from one end tower to the other. This is easily proved through demonstration. - Roderick Kimball, Nov 22 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A103452.

Essentially identical to A024023.

[0^n+3^n-1: n in [0..30] ]; // Vincenzo Librandi, Apr 30 2011

Table[If[n==0, 1, 3^n -1], {n, 0, 30}] (* G. C. Greubel, Jun 18 2021 *)
LinearRecurrence[{4,-3},{1,2,8},30] (* Harvey P. Dale, Feb 13 2022 *)

a(n) = if(n==0, 1, 3^n-1); \\ Altug Alkan, Nov 22 2015

[3^n -1 +0^n for n in (0..30)] # G. C. Greubel, Jun 18 2021

G.f.: (1 -2*x +3*x^2)/((1-x)*(1-3*x)).
a(n) = Sum_{k=0..n} A103452(n, k)*3^k.
a(n) = Sum_{k=0..n} (2*0^(n-k) - 1)*0^(k*(n-k))*3^k.
From G. C. Greubel, Jun 18 2021: (Start)
E.g.f.: 1 - exp(x) + exp(3*x).
a(n) = [n=0] + 2*A003462(n). (End)

A103456 a(n) = 0^n + 10^n - 1.

Original entry on oeis.org

1, 9, 99, 999, 9999, 99999, 999999, 9999999, 99999999, 999999999, 9999999999, 99999999999, 999999999999, 9999999999999, 99999999999999, 999999999999999, 9999999999999999, 99999999999999999, 999999999999999999

Offset: 0

Paul Barry, Feb 06 2005

A transform of 10^n under the matrix A103452.

Except for n = 0, the same as A002283. - Felix Fröhlich, Jun 22 2021

G. C. Greubel, Table of n, a(n) for n = 0..450
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002283, A103452.

[1] cat [10^n -1: n in [1..40]]; // G. C. Greubel, Jun 21 2021

Table[Boole[n==0] + 10^n -1, {n,0,40}] (* Alonso del Arte, Nov 03 2019 *)

a(n) = 0^n + 10^n - 1 \\ Felix Fröhlich, Jun 22 2021

Vec((1 - 2*x + 10*x^2)/((1 - x)*(1 - 10*x)) + O(x^20)) \\ Felix Fröhlich, Jun 22 2021

[1]+[10^n -1 for n in (1..40)] # G. C. Greubel, Jun 21 2021

G.f.: (1 - 2*x + 10*x^2)/((1 - x)*(1 - 10*x));
a(n) = Sum_{k = 0..n} A103452(n, k)*10^k;
a(n) = Sum_{k = 0..n} (2*0^(n-k) - 1)*0^(k*(n-k))*10^k.
a(n) = A002283(n), n > 0. - R. J. Mathar, Aug 30 2008
E.g.f.: 1 - exp(x) + exp(10*x). - G. C. Greubel, Jun 21 2021

cp's OEIS Frontend

A103451 Triangular array T read by rows: T(n, 0) = T(n, n) = 1, T(n, k) = 0 for 0 < k < n.

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A103454 a(n) = 0^n + 4^n - 1.

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A206735 Triangle T(n,k), read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A103455 a(n) = 0^n + 5^n - 1.

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Magma

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A103453 a(n) = 0^n + 3^n - 1.

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Programs

Magma

Mathematica

PARI

Sage

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A103456 a(n) = 0^n + 10^n - 1.

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Magma

Mathematica

PARI

PARI

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