A040027 -id:A040027 - cp's OEIS frontend

A186020 Eigentriangle of the binomial matrix.

1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 15, 9, 4, 1, 1, 52, 31, 14, 5, 1, 1, 203, 121, 54, 20, 6, 1, 1, 877, 523, 233, 85, 27, 7, 1, 1, 4140, 2469, 1101, 400, 125, 35, 8, 1, 1, 21147, 12611, 5625, 2046, 635, 175, 44, 9, 1, 1, 115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1, 1

Offset: 0

Paul Barry, Feb 10 2011

Reversal of Gould triangle A121207. First column is A000110. Second column is A040027.
Row sums are A186021. Diagonal sums are A186022.
Construction is described by Paul D. Hanna in A121207. The method of construction is general for this class of eigentriangle.

			Triangle T begins
       1;
       1,     1;
       2,     1,     1;
       5,     3,     1,     1;
      15,     9,     4,     1,    1;
      52,    31,    14,     5,    1,   1;
     203,   121,    54,    20,    6,   1,   1;
     877,   523,   233,    85,   27,   7,   1,  1;
    4140,  2469,  1101,   400,  125,  35,   8,  1,  1;
   21147, 12611,  5625,  2046,  635, 175,  44,  9,  1, 1;
  115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1, 1;
Inverse is the identity matrix I minus binomial matrix B shifted down once, or
T^{-1}(n,k)=if(k=n,1,if(k<n,-binomial(n-1,k),0)). This begins
   1;
  -1,  1;
  -1, -1,   1;
  -1, -2,  -1,   1;
  -1, -3,  -3,  -1,   1;
  -1, -4,  -6,  -4,  -1,   1;
  -1, -5, -10, -10,  -5,  -1,   1;
  -1, -6, -15, -20, -15,  -6,  -1,  1;
  -1, -7, -21, -35, -35, -21,  -7, -1,  1;
  -1, -8, -28, -56, -70, -56, -28, -8, -1, 1;
Production matrix is
      1,     1;
      1,     0,    1;
      2,     1,    0,    1;
      5,     3,    1,    0,   1;
     15,     9,    4,    1,   0,   1;
     52,    31,   14,    5,   1,   0,  1;
    203,   121,   54,   20,   6,   1,  0, 1;
    877,   523,  233,   85,  27,   7,  1, 0, 1;
   4140,  2469, 1101,  400, 125,  35,  8, 1, 0, 1;
  21147, 12611, 5625, 2046, 635, 175, 44, 9, 1, 0, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv:1107.5490 [math.CO], 2011.
Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Cf. A000110, A007318, A121207, A124496, A160185, A186021, A186022.

t[n_, k_] := t[n, k] = If[k == 0, 1, Sum[t[n-j, k-j] Binomial[n-1, j-1], {j, 1, k}]];
T[n_, k_] := t[n, n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 27 2018 *)

Lower triangular (infinite) matrix T = (U - D*P)^{-1} with the unit matrix U, the Pascal matrix P from A007318 and the matrix D with elements delta_{i,j+1}, for i, j >= 0 (row 0 has only 0s). From the Paul Barry paper rewritten in matrix notation. T satisfies P*T = D'*(T - U), with D' the transposed matrix D, that is the diagonal of T has been erased and the row index shifted on the r.h.s. (showing that the name Eigentriangle or -matrix is a misnomer). For finite N X N matrices P*T = D'*(T - U), only up to the last row. - Wolfdieter Lang, Apr 07 2021

A045500 Fifth-from-right diagonal of triangle A121207.

Original entry on oeis.org

1, 1, 6, 27, 125, 635, 3488, 20425, 126817, 831915, 5744784, 41618459, 315388311, 2493721645, 20526285716, 175529425815, 1556577220651, 14290644428279, 135624265589086, 1328702240382589, 13420603191219111, 139592874355534071

Offset: 0

Henry Gould

With leading 0 and offset 4: number of permutations beginning with 54321 and avoiding 1-23. - Ralf Stephan, Apr 25 2004

a(n) is the number of set partitions of {1,2,...,n+4} in which the last block has length 4: the blocks are arranged in order of their least element. - Don Knuth, Jun 12 2017

See also references under sequence A040027.

Seiichi Manyama, Table of n, a(n) for n = 0..573
S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2014.

Cf. A040027, A045499, A045501, A121207, A346060.

Column k=4 of A124496.

a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n+3, k+4]*a[k], {k, 0, n-1}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 14 2018, after Vladeta Jovovic *)

{a(n)=local(A=1+x); for(i=1, n, A=1+x*subst(A, x, x/(1-x+x*O(x^n)))/(1-x)^5); polcoeff(A, n)} /* Paul D. Hanna, Mar 23 2012 */

# The function Gould_diag is defined in A121207.
A045500_list = lambda size: Gould_diag(5, size)
print(A045500_list(24)) # Peter Luschny, Apr 24 2016

a(n+1) = Sum_{k=0..n} binomial(n+4, k+4)*a(k). - Vladeta Jovovic, Nov 10 2003
With offset 4, e.g.f.: x^4 + exp(exp(x))/24 * int[0..x, t^4*exp(-exp(t)+t) dt]. - Ralf Stephan, Apr 25 2004
O.g.f. satisfies: A(x) = 1 + x*A( x/(1-x) ) / (1-x)^5. - Paul D. Hanna, Mar 23 2012

More terms from Vladeta Jovovic, Nov 10 2003

Entry revised by N. J. A. Sloane, Dec 11 2006

A045501 Third-from-right diagonal of triangle A121207.

Original entry on oeis.org

1, 1, 4, 14, 54, 233, 1101, 5625, 30846, 180474, 1120666, 7352471, 50772653, 367819093, 2787354668, 22039186530, 181408823710, 1551307538185, 13756835638385, 126298933271289, 1198630386463990, 11742905240821910

Offset: 1

Henry Gould

With leading 0 and offset 2: number of permutations beginning with 321 and avoiding 1-23. - Ralf Stephan, Apr 25 2004
Second diagonal in table of binomial recurrence coefficients. Related to A040027. - Vladeta Jovovic, Feb 05 2008
Equals eigensequence of triangle A104712. - Gary W. Adamson, Apr 10 2009
a(n) is the number of set partitions of {1,2,...,n+1} in which the last block has length 2; the blocks are arranged in order of their least element. - Don Knuth, Jun 12 2017

S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.

Cf. A045499, A045500.
Cf. A104712. - Gary W. Adamson, Apr 10 2009
Column k=2 of A124496.

a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n, k+1]*a[k], {k, 0, n-1}];
Array[a, 22] (* Jean-François Alcover, Jul 14 2018, after Vladeta Jovovic *)

{a(n)=local(A=x+x^2); for(i=1, n, A=x+x*subst(A, x, x/(1-x+x*O(x^n)))/(1-x)^2); polcoeff(A, n)} /* Paul D. Hanna, Mar 23 2012 */

# The function Gould_diag is defined in A121207.
A045501_list = lambda size: Gould_diag(3, size)
print(A045501_list(24)) # Peter Luschny, Apr 24 2016

a(n+1) = Sum_{k=0..n} binomial(n+2, k+2)*a(k). - Vladeta Jovovic, Nov 10 2003
With offset 2, e.g.f.: x^2 + exp(exp(x))/2 * Integral_{0..x} t^2*exp(-exp(t)+t) dt. - Ralf Stephan, Apr 25 2004
G.f.: A(x) = Sum_{k>=0} x^(k+1)/((1-k*x)^2 * Product_{m=0..k} (1 - m*x)). - Vladeta Jovovic, Feb 05 2008
O.g.f. satisfies: A(x) = x + x*A( x/(1-x) ) / (1-x)^2. - Paul D. Hanna, Mar 23 2012

More terms from Vladeta Jovovic, Nov 10 2003

Entry revised by N. J. A. Sloane, Dec 11 2006

A362925 Triangle read by rows: T(n,m), n >= 0, 0 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 4, 1, 15, 15, 13, 8, 1, 52, 52, 47, 35, 16, 1, 203, 203, 188, 153, 97, 32, 1, 877, 877, 825, 706, 515, 275, 64, 1, 4140, 4140, 3937, 3479, 2744, 1785, 793, 128, 1, 21147, 21147, 20270, 18313, 15177, 11002, 6347, 2315, 256, 1, 115975, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1

Offset: 0

N. J. A. Sloane, Aug 10 2023, based on an email from Don Knuth

A variant of A113547 and A362924. See those entries for further information.

			Triangle begins:
       1;
       1,      1;
       2,      2,      1;
       5,      5,      4,      1;
      15,     15,     13,      8,     1;
      52,     52,     47,     35,    16,     1;
     203,    203,    188,    153,    97,    32,     1;
     877,    877,    825,    706,   515,   275,    64,     1;
    4140,   4140,   3937,   3479,  2744,  1785,   793,   128,    1;
   21147,  21147,  20270,  18313, 15177, 11002,  6347,  2315,  256,   1;
  115975, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Row sums are A000110(n+1).
Columns k=0+1,2-5 give A000110, A078468(n-2) (for n>=2), A383052(n-3) (for n>=3), A383053(n-4) (for n>=4), A383054(n-5) (for n>=5).
T(n+j,n) give (for j=0-2): A000012, A000079, A007689.
T(2n,n) gives A367820.
Cf. A040027, A113547, A362924.

T:= (n, k)-> add(Stirling2(n-k, j)*(j+1)^k, j=0..n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 01 2023

A362925[n_, m_]:=Sum[StirlingS2[n-m,k](k+1)^m,{k,0,n-m}];
Table[A362925[n,m],{n,0,15},{m,0,n}] (* Paolo Xausa, Dec 04 2023 *)

Sum_{k=0..n} (k+1) * T(n,k) = A040027(n+1). - Alois P. Heinz, Dec 02 2023

A343523 a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 2, 8, 34, 164, 878, 5136, 32490, 220476, 1594470, 12223016, 98876322, 840804820, 7491247006, 69730182720, 676390547034, 6821988655468, 71398971351510, 774032400213336, 8677733804696594, 100459693769214980, 1199306075189097230, 14746332963835756400, 186534818943430728906

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Georg Fischer, Table of n, a(n) for n = 0..400

Cf. A001861, A004123, A035009, A040027, A343975, A344735, A344840, A345077, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 23}]
nmax = 23; A[] = 0; Do[A[x] = 1 + 2 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 2 * x * A(x/(1 - x)) / (1 - x)^2.

A351437 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - x)) / (1 - x)^2.

Original entry on oeis.org

1, 1, 1, 3, 7, 17, 47, 145, 481, 1691, 6295, 24805, 103095, 449805, 2052081, 9762699, 48334855, 248568321, 1325297879, 7312927481, 41694974649, 245288605059, 1487041552343, 9279329735685, 59537092965663, 392371097100373, 2653606218921673, 18400405626141667, 130712743774279015

Offset: 0

Ilya Gutkovskiy, Feb 11 2022

nonn

Cf. A007476, A040027, A351438.

nmax = 28; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k + 1] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 28}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-1,k+1) * a(k).

A046936 Same rule as Aitken triangle (A011971) except a(0,0)=0, a(1,0)=1.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 6, 9, 9, 12, 16, 22, 31, 31, 40, 52, 68, 90, 121, 121, 152, 192, 244, 312, 402, 523, 523, 644, 796, 988, 1232, 1544, 1946, 2469, 2469, 2992, 3636, 4432, 5420, 6652, 8196, 10142, 12611, 12611, 15080, 18072, 21708, 26140

Offset: 0

N. J. A. Sloane, R. K. Guy

			Triangle starts:
0,
1, 1,
1, 2, 3,
3, 4, 6, 9,
9, 12, 16, 22, 31,
31, 40, 52, 68, 90, 121,
121, 152, 192, 244, 312, 402, 523,
523, 644, 796, 988, 1232, 1544, 1946, 2469,
2469, 2992, 3636, 4432, 5420, 6652, 8196, 10142, 12611,
12611, 15080, ...

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
Don Knuth, Email to N. J. A. Sloane, Jan 29 2018

Borders give A040027. Reading across rows gives A007604.

Cf. A121207, A298804.

a046936 n k = a046936_tabl !! n !! k
a046936_row n = a046936_tabl !! n
a046936_tabl = [0] : iterate (\row -> scanl (+) (last row) row) [1,1]
-- Reinhard Zumkeller, Jan 01 2014

a[0, 0] = 0; a[1, 0] = 1; a[n_, 0] := a[n, 0] = a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 15 2013 *)

from itertools import accumulate
def A046936(): # Compare function Gould_diag in A121207.
    yield [0]
    accu = [1, 1]
    while True:
        yield accu
        accu = list(accumulate([accu[-1]] + accu))
g = A046936()
[next(g) for  in range(9)] # _Peter Luschny, Apr 25 2016

A343975 a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 3, 15, 81, 489, 3237, 23211, 178707, 1467051, 12768345, 117263829, 1131901521, 11444383251, 120847326879, 1329303053391, 15197269729689, 180211641841353, 2212525627591533, 28078380387448515, 367782119667874083, 4965441830591976339, 69014083524412401873, 986364827548578356421

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A027710, A032033, A040027, A078940, A343523, A344735, A344840, A345077, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1 + 3 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 3 * x * A(x/(1 - x)) / (1 - x)^2.

A344735 a(0) = 1; a(n) = 4 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 4, 24, 156, 1120, 8740, 73384, 657900, 6259184, 62876852, 664134968, 7349666684, 84956020864, 1023006054980, 12802727760840, 166174971580684, 2232866214809360, 31007771007956948, 444360490882720344, 6562410784684023452, 99749853821538893216, 1558780425524233360740

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A040027, A078944, A078945, A094417, A343523, A343975, A344840, A345077, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 4 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1 + 4 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 4 * x * A(x/(1 - x)) / (1 - x)^2.

A344840 a(0) = 1; a(n) = 5 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 5, 35, 265, 2195, 19625, 187755, 1909185, 20521515, 232124745, 2752591475, 34108980105, 440444019835, 5912197332865, 82320781521195, 1186703083508025, 17680850448587155, 271845880552898985, 4307188044378111915, 70236616096770062945, 1177406236243423738475

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A040027, A094418, A144180, A343523, A343975, A344735, A345077, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 5 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1 + 5 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 5 * x * A(x/(1 - x)) / (1 - x)^2.

cp's OEIS Frontend

A186020 Eigentriangle of the binomial matrix.

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A045500 Fifth-from-right diagonal of triangle A121207.

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A045501 Third-from-right diagonal of triangle A121207.

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A362925 Triangle read by rows: T(n,m), n >= 0, 0 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.

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A343523 a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(n,k) * a(k-1).

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A351437 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - x)) / (1 - x)^2.

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A046936 Same rule as Aitken triangle (A011971) except a(0,0)=0, a(1,0)=1.

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A343975 a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(k-1).