A091613 -id:A091613 - cp's OEIS frontend

A091614 Matrix inverse of triangle A091613.

1, -1, 1, -3, 0, 1, -1, -3, 0, 1, 5, -6, -2, 0, 1, 13, -4, -5, -2, 0, 1, 27, 1, -7, -4, -2, 0, 1, 41, 12, -4, -6, -4, -2, 0, 1, 43, 35, 4, -6, -5, -4, -2, 0, 1, 25, 72, 18, 0, -5, -5, -4, -2, 0, 1, -23, 128, 40, 11, -2, -4, -5, -4, -2, 0, 1, -157, 205, 77, 30, 8, -1, -4, -5, -4, -2, 0, 1

Offset: 1

Christian G. Bower, Jan 23 2004

			Triangle begins as:
    1;
   -1,   1;
   -3,   0,  1;
   -1,  -3,  0,  1;
    5,  -6, -2,  0,  1;
   13,  -4, -5, -2,  0,  1;
   27,   1, -7, -4, -2,  0,  1;
   41,  12, -4, -6, -4, -2,  0,  1;
   43,  35,  4, -6, -5, -4, -2,  0,  1;
   25,  72, 18,  0, -5, -5, -4, -2,  0,  1;
  -23, 128, 40, 11, -2, -4, -5, -4, -2,  0,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A091613, A091623 (first column).

b[n_, l_, k_]:= b[n, l, k]= If[n==0, 1, Sum[If[i==l, 0, Sum[b[n-i*j, i, k], {j, Min[k, n/i]}]], {i, n}]];
t[n_, k_]:= b[n, 0, k] - b[n, 0, k-1]; (* t = A091613 *)
M:= With[{p = 16}, Table[t[n, k], {n, p}, {k, p}]];
T:= Inverse[M];
Table[T[[n, k]], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Nov 27 2021 *)

Name corrected by G. C. Greubel, Nov 27 2021

A091615 Matrix square of triangle A091613.

Original entry on oeis.org

1, 2, 1, 6, 0, 1, 11, 6, 0, 1, 26, 12, 4, 0, 1, 61, 26, 10, 4, 0, 1, 132, 70, 26, 8, 4, 0, 1, 290, 174, 62, 24, 8, 4, 0, 1, 638, 404, 156, 60, 22, 8, 4, 0, 1, 1390, 946, 385, 146, 58, 22, 8, 4, 0, 1, 3012, 2192, 921, 362, 144, 56, 22, 8, 4, 0, 1, 6496, 5008, 2177, 876, 350, 142

Offset: 1

Christian G. Bower, Jan 23 2004

			1; 2,1; 6,0,1; 11,6,0,1; 26,12,4,0,1; ...

Row sums: A091621. Column 1: A091622.

A034007 First differences of A045891.

Original entry on oeis.org

1, 0, 2, 4, 9, 20, 44, 96, 208, 448, 960, 2048, 4352, 9216, 19456, 40960, 86016, 180224, 376832, 786432, 1638400, 3407872, 7077888, 14680064, 30408704, 62914560, 130023424, 268435456, 553648128, 1140850688, 2348810240

Offset: 0

Wolfdieter Lang

Let M_n be the n X n matrix m_(i,j) = 4 + abs(i-j) then det(M_n) = (-1)^(n+1)*a(n+2). - Benoit Cloitre, May 28 2002
Number of ordered pairs of (possibly empty) ordered partitions, each not beginning with 1. - Christian G. Bower, Jan 23 2004
If X_1, X_2, ..., X_n are 2-blocks of a (2n+4)-set X then, for n>=1, a(n+3) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S^2)^2; see A291000. - Clark Kimberling, Aug 24 2017
Conjecture 1: For compositions of n+k-1, a(n) is the number of runs of 1 of length k. Example: Among the compositions of 4+2-1 = 5, there are a(4) = 4 runs of two 1's: 3,[1,1]; [1,1],3; 1,2,[1,1] and [1,1],2,1. - Gregory L. Simay, Feb 18 2018
Conjecture 2: More generally, let R(n,m,k) = the number of runs of k m's in all compositions of n. Then R(n,m,k) = A045623(n-m*k) - 2*A045623(n-m*(k+1)) + A045623(n-m*(k+2)). For example, R(7,1,1) = A045623(6) - 2*A045623(5) + A045623(4) = 144 - 2*64 + 28 = 44 = a(7). - Gregory L. Simay, Feb 20 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Frank Ellermann, Illustration of binomial transforms.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A045623, A045891, A291000.
Convolution of A034008 with itself.
Columns of A091613 converge to this sequence.

[1,0,2] cat [(n+5)*2^(n-4): n in [3..30]]; // G. C. Greubel, Sep 27 2022

Join[{1,0,2,a=4},Table[a=(2*(n+7)*a)/(n+6),{n,2,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)

a(n)=if(n<3,[1,0,2][n+1],(n+5)*2^(n-4)) \\ Charles R Greathouse IV, Jun 01 2011

[1,0,2]+[(n+5)*2^(n-4) for n in range(3,30)] # G. C. Greubel, Sep 27 2022

a(n) = Sum_{k = 0..n-3} (k+4)*binomial(n-3,k) for n >= 3. - N. J. A. Sloane, Jan 30 2008
a(n) = (n+5)*2^(n-4), n >= 3; a(0)=1, a(1)=0, a(2)=2.
G.f.: ((1-x)^2/(1-2*x))^2.
a(n) = Sum_{k=0..n} (k+1)*C(n-3,n-k). - Peter Luschny, Apr 20 2015
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=2} 1/a(n) = 512*log(2) - 74327/210.
Sum_{n>=2} (-1)^n/a(n) = 14579/70 - 512*log(3/2). (End)
E.g.f.: (1/16)*(11 - 12*x + 2*x^2 + (5+2*x)*exp(2*x)). - G. C. Greubel, Sep 27 2022

A091616 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 2 times and no part is repeated consecutively more than 2 times.

Original entry on oeis.org

1, 0, 3, 6, 10, 23, 50, 99, 200, 404, 805, 1599, 3166, 6225, 12223, 23934, 46713, 90995, 176935, 343395, 665474, 1287918, 2489467, 4806805, 9272272, 17870317, 34414163, 66226890, 127365537, 244803475, 470278815, 902997083, 1733124564, 3325087228, 6377076320

Offset: 2

Christian G. Bower, Jan 23 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Column k=2 of A091613.

Cf. A128695.

b:= proc(n, l, k) option remember; `if`(n=0, 1, add(`if`(
      i=l, 0, add(b(n-i*j, i, k), j=1..min(k, n/i))), i=1..n))
    end:
a:= n-> b(n, 0, 2) -b(n, 0, 1):
seq(a(n), n=2..50);  # Alois P. Heinz, Feb 08 2017

b[n_, l_, k_] := b[n, l, k] = If[n == 0, 1, Sum[If[i == l, 0, Sum[b[n - i*j, i, k], {j, 1, Min[k, n/i]}]], {i, 1, n}]];
a[n_] := b[n, 0, 2] - b[n, 0, 1];
Table[a[n], {n, 2, 50}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)
nmax = 50; Drop[CoefficientList[Series[1/(1 - Sum[(x^k + x^(2*k))/(1 + x^k + x^(2*k)), {k, 1, nmax}]) - 1/(1 - Sum[x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x], 2] (* Vaclav Kotesovec, Jul 07 2020 *)

a(n) ~ c * d^n, where d = 1.9107639262818041675000243699745706859... (see A128695), c = 0.499300813712837808621944870186032611... - Vaclav Kotesovec, Sep 21 2019

a(n) = A128695(n) - A003242(n). - Vaclav Kotesovec, Jul 07 2020

A091617 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 3 times and no part is repeated consecutively more than 3 times.

Original entry on oeis.org

1, 0, 2, 5, 11, 22, 48, 105, 223, 468, 979, 2037, 4224, 8710, 17906, 36693, 74973, 152795, 310669, 630353, 1276544, 2580614, 5208522, 10496954, 21126102, 42464498, 85255651, 170980512, 342553389, 685641724, 1371134291, 2739685107, 5469923466, 10912945300

Offset: 3

Christian G. Bower, Jan 23 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A091613.

b:= proc(n, l, k) option remember; `if`(n=0, 1, add(`if`(
      i=l, 0, add(b(n-i*j, i, k), j=1..min(k, n/i))), i=1..n))
    end:
a:= n-> b(n, 0, 3) -b(n, 0, 2):
seq(a(n), n=3..50);  # Alois P. Heinz, Feb 08 2017

b[n_, l_, k_] := b[n, l, k] = If[n == 0, 1, Sum[If[i == l, 0, Sum[b[n - i*j, i, k], {j, 1, Min[k, n/i]}]], {i, 1, n}]];
a[n_] := b[n, 0, 3] - b[n, 0, 2];
Table[a[n], {n, 3, 50}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)

A091618 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 4 times and no part is repeated consecutively more than 4 times.

Original entry on oeis.org

1, 0, 2, 4, 10, 22, 46, 101, 218, 466, 991, 2093, 4405, 9232, 19288, 40169, 83416, 172806, 357170, 736710, 1516714, 3117133, 6396116, 13105012, 26814264, 54795330, 111842771, 228030558, 464439943, 945029466, 1921169854, 3902239742, 7919743405, 16061152291

Offset: 4

Christian G. Bower, Jan 23 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A091613.

b:= proc(n, l, k) option remember; `if`(n=0, 1, add(`if`(
      i=l, 0, add(b(n-i*j, i, k), j=1..min(k, n/i))), i=1..n))
    end:
a:= n-> b(n, 0, 4) -b(n, 0, 3):
seq(a(n), n=4..50);  # Alois P. Heinz, Feb 08 2017

b[n_, l_, k_] := b[n, l, k] = If[n == 0, 1, Sum[If[i == l, 0, Sum[b[n - i*j, i, k], {j, 1, Min[k, n/i]}]], {i, 1, n}]];
a[n_] :=  b[n, 0, 4] - b[n, 0, 3];
Table[a[n], {n, 4, 50}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)

A091619 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 5 times and no part is repeated consecutively more than 5 times.

Original entry on oeis.org

1, 0, 2, 4, 9, 21, 46, 98, 213, 459, 979, 2082, 4408, 9294, 19535, 40937, 85561, 178392, 371131, 770556, 1596936, 3303986, 6825163, 14078718, 29002331, 59670920, 122627798, 251735331, 516247636, 1057687986, 2165046327, 4428017780, 9049083763, 18478717262

Offset: 5

Christian G. Bower, Jan 23 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=5 of A091613.

b:= proc(n, l, k) option remember; `if`(n=0, 1, add(`if`(
      i=l, 0, add(b(n-i*j, i, k), j=1..min(k, n/i))), i=1..n))
    end:
a:= n-> b(n, 0, 5) -b(n, 0, 4):
seq(a(n), n=5..50);  # Alois P. Heinz, Feb 08 2017

b[n_, l_, k_] := b[n, l, k] = If[n == 0, 1, Sum[If[i == l, 0, Sum[b[n - i*j, i, k], {j, 1, Min[k, n/i]}]], {i, 1, n}]];
a[n_] := b[n, 0, 5] - b[n, 0, 4];
Table[a[n], {n, 5, 50}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)

A091620 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 6 times and no part is repeated consecutively more than 6 times.

Original entry on oeis.org

1, 0, 2, 4, 9, 20, 45, 98, 210, 453, 971, 2068, 4387, 9275, 19545, 41064, 86055, 179913, 375338, 781497, 1624250, 3370238, 6982398, 14445576, 29846586, 61591860, 126956859, 261411737, 537723480, 1105055809, 2268948882, 4654815069, 9541957646, 19545570684

Offset: 6

Christian G. Bower, Jan 23 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A091613.

b:= proc(n, l, k) option remember; `if`(n=0, 1, add(`if`(
      i=l, 0, add(b(n-i*j, i, k), j=1..min(k, n/i))), i=1..n))
    end:
a:= n-> b(n, 0, 6) -b(n, 0, 5):
seq(a(n), n=6..50);  # Alois P. Heinz, Feb 08 2017

b[n_, l_, k_] := b[n, l, k] = If[n == 0, 1, Sum[If[i == l, 0, Sum[b[n - i*j, i, k], {j, 1, Min[k, n/i]}]], {i, 1, n}]];
a[n_] := b[n, 0, 6] - b[n, 0, 5];
Table[a[n], {n, 6, 50}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)

A091623 Column 1 of triangle A091614.

Original entry on oeis.org

1, -1, -3, -1, 5, 13, 27, 41, 43, 25, -23, -157, -447, -1011, -2087, -4081, -7685, -14141, -25707, -46251, -82729, -147295, -261317, -462403, -816653, -1439457, -2532779, -4448035, -7794637, -13624815, -23744693, -41230561, -71277787

Offset: 1

Christian G. Bower, Jan 23 2004

sign

G. C. Greubel, Table of n, a(n) for n = 1..120

Cf. A091614.

b[n_, l_, k_]:= b[n, l, k]= If[n==0, 1, Sum[If[i==l, 0, Sum[b[n-i*j, i, k], {j, 1, Min[k, n/i]}]], {i, n}]];
t[n_, k_]:= t[n, k]= If[k>n, 0, b[n, 0, k] - b[n, 0, k-1]]; (* A091613 *)
M:= With[{p=53}, Table[t[n, k], {n, p}, {k, p}]];
T := Inverse[M]; (* A091614 *)
Table[T[[n, 1]], {n, 50}] (* G. C. Greubel, Nov 27 2021 *)

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A091614 Matrix inverse of triangle A091613.

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A091615 Matrix square of triangle A091613.

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A034007 First differences of A045891.

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A091616 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 2 times and no part is repeated consecutively more than 2 times.

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A091617 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 3 times and no part is repeated consecutively more than 3 times.

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A091618 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 4 times and no part is repeated consecutively more than 4 times.

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A091619 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 5 times and no part is repeated consecutively more than 5 times.

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A091620 Number of compositions (ordered partitions) of n such that some part is repeated consecutively 6 times and no part is repeated consecutively more than 6 times.

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A091623 Column 1 of triangle A091614.

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