A006351 -id:A006351 - cp's OEIS frontend

A379460 Array read by downward antidiagonals: A(n,k) = A(n,k-1) + (k+1)*(A(n-1,k) + A(n-1,k+1)) with A(n,0) = A(n-1,0) + A(n-1,1), A(0,k) = 1, n >= 0, k >= 0.

1, 1, 2, 1, 6, 8, 1, 12, 44, 52, 1, 20, 140, 420, 472, 1, 30, 340, 1860, 5032, 5504, 1, 42, 700, 6020, 28672, 72912, 78416, 1, 56, 1288, 15960, 116592, 508704, 1241648, 1320064, 1, 72, 2184, 36792, 380352, 2496480, 10257200, 24317760, 25637824, 1, 90, 3480, 76440, 1059744, 9696960, 59030960, 232182240, 538637824, 564275648

Offset: 0

Mikhail Kurkov, Dec 23 2024

			Array begins:
=============================================================
n\k|     0       1        2        3         4          5 ...
---+---------------------------------------------------------
0  |     1       1        1        1         1          1 ...
1  |     2       6       12       20        30         42 ...
2  |     8      44      140      340       700       1288 ...
3  |    52     420     1860     6020     15960      36792 ...
4  |   472    5032    28672   116592    380352    1059744 ...
5  |  5504   72912   508704  2496480   9696960   31778208 ...
  ...

Cf. A006351.

A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, 1)); r[1] = v[1..n+1];
for(i=1, m, v[1] = v[1] + v[2]; for(k=2, #v-1, v[k] = v[k-1] + k*(v[k] + v[k+1])); r[1+i] = v[1..n+1]); Mat(r)}
{ A(5) }

Conjecture: A(n,0) = A006351(n+1).

A298673 Inverse matrix of A135494.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 26, 19, 6, 1, 236, 170, 55, 10, 1, 2752, 1966, 645, 125, 15, 1, 39208, 27860, 9226, 1855, 245, 21, 1, 660032, 467244, 155764, 32081, 4480, 434, 28, 1, 12818912, 9049584, 3031876, 635124, 92001, 9576, 714, 36, 1, 282137824, 198754016, 66845340, 14180440, 2108085, 230097, 18690, 1110, 45, 1

Offset: 1

Tom Copeland, Jan 24 2018

Since this is the inverse matrix of A135494 with row polynomials q_n(t), first introduced in that entry by R. J. Mathar, and the row polynomials p_n(t) of this entry are a binomial Sheffer polynomial sequence, the row polynomials of the inverse pair are umbral compositional inverses, i.e., p_n(q.(t)) = q_n(p.(t)) = t^n. For example, p_3(q.(t)) = 4q_1(t) + 3q_2(t) + q_3(t) = 4t + 3(-t + t^2) + (-t -3t^2 +t^3) = t^3. In addition, both sequences possess the umbral convolution property (p.x) + p.(y))^n = p_n(x+y) with p_0(t) = 1.

This is the inverse of the Bell matrix generated by A153881; for the definition of the Bell matrix see the link. - Peter Luschny, Jan 26 2018

			Matrix begins as
     1;
     1;    1;
     4,    3,    1;
    26,   19,    6,    1;
   236,  170,   55,   10,    1;
  2752, 1966,  645,  125,   15,    1;

Peter Luschny, The Bell transform
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.

Cf. A000311, A000169, A000670, A006351, A135494.

# The function BellMatrix is defined in A264428. Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n=0, 1, -1), 9): MatrixInverse(%); # Peter Luschny, Jan 26 2018

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, If[n == 0, 1, -1]], rows = 12] // Inverse;
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

E.g.f.: e^[p.(t)x] = e^[t*h(x)] = exp[t*[(x-1)/2 + T{ (1/2) * exp[(x-1)/2] }], where T is the tree function of A000169 related to the Lambert function. h(x) = sum(j=1,...) A000311(j) * x^j / j! = exp[xp.'(0)], so the first column of this entry's matrix is A000311(n) for n > 0 and the second column of the full matrix for p_n(t) to n >= 0. The compositional inverse of h(x) is h^(-1)(x) = 1 + 2x - e^x.
The lowering operator is L = h^(-1)(D) = 1 + 2D - e^D with D = d/dt, i.e., L p_n(t) = n * p_(n-1)(t). For example, L p_3(t) = (D - D^2! - D^3/3! - ...) (4t + 6t^ + t^3) = 3 (t + t^2) = 3 p_2(t).
The raising operator is R = t * 1/[d[h^(-1)(D)]/dD] = t * 1/[2 - e^D)] = t (1 + D + 3D^2/2! + 13D^3/3! + ...). The coefficients of R are A000670. For example, R p_2(t) = t (1 + D + 3D^2/2!  + ...)  (t + t^2) = 4t + 3t^2 + t^3   = p_3(t).
The row sums are A006351, or essentially 2*A000311.
Conjectures from Mikhail Kurkov, Mar 01 2025: (Start)
T(n,k) = Sum_{j=0..n-k} binomial(n+j-1, k-1)*A269939(n-k, j) for 1 <= k <= n.
T(n,k) = A(n-1,k,0) for n > 0, k > 0 where A(n,k,q) = A(n-1,k,q+1) + 2*(q+1)!*Sum_{j=0..q} A(n-1,k,j)/j! for n >= 0, k > 0, q >= 0 with A(0,k,q) = Stirling1(q+1,k) for k > 0, q >= 0 (see A379458). In other words, T(n,k) = Sum_{j=0}^{n-1} A379460(n-j-1,j)*Stirling1(j+1,k) for n > 0, k > 0.
Recursion for the n-th row (independently of other rows): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} b(j-1)*binomial(-k,j)*T(n,k+j-1)*(-1)^j for 1 <= k < n with T(n,n) = 1 where b(n) = 1 + 4*Sum_{i=1..n} A135148(i).
Recursion for the k-th column (independently of other columns): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} c(j-1)*binomial(n,j)*T(n-j+1,k) for 1 <= k < n with T(n,n) = 1 where c(n) = A000311(n+1) + (n-1)*A000311(n). (End)

A379459 Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + 2(k+1)!Sum_{j=0..k} A(n-1,j)/j! with A(0,k) = 2*(k+1)!, n >= 0, k >= 0.

Original entry on oeis.org

2, 4, 8, 12, 36, 52, 48, 192, 368, 472, 240, 1200, 2880, 4560, 5504, 1440, 8640, 24960, 47280, 67408, 78416, 10080, 70560, 238560, 527520, 871584, 1163232, 1320064, 80640, 645120, 2499840, 6330240, 11926656, 18031104, 22997696, 25637824, 725760, 6531840, 28546560, 81527040, 172811520, 292642560, 415728960, 513000000, 564275648

Offset: 0

Mikhail Kurkov, Dec 23 2024

			Array begins:
================================================================
n\k|     0       1        2         3          4           5 ...
---+------------------------------------------------------------
0  |     2       4       12        48        240        1440 ...
1  |     8      36      192      1200       8640       70560 ...
2  |    52     368     2880     24960     238560     2499840 ...
3  |   472    4560    47280    527520    6330240    81527040 ...
4  |  5504   67408   871584  11926656  172811520  2649749760 ...
5  | 78416 1163232 18031104 292642560 4977020160 88700451840 ...
  ...

Cf. A006351.

A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, 2*k!)); r[1] = v[1..n+1];
for(i=1, m, v=vector(#v-1, k, v[k+1] + 2*k!*sum(j=1, k, v[j]/(j-1)!)); r[1+i] = v[1..n+1]); Mat(r)}
{ A(5) }

Conjecture: A(n,0) = A006351(n+2).

A232005 Number of distinct resistances that can be produced from a circuit of resistors with resistances 1, 2, ..., n using only series and parallel combinations.

Original entry on oeis.org

1, 2, 8, 48, 386, 3781, 49475, 762869, 13554897, 266817541

Offset: 1

Dave R.M. Langers, Nov 16 2013

Found by exhaustive search: all configurations of resistors were enumerated, resistances calculated, sorted, and distinct values counted.
This sequence allows any circuits to be combined in series or in parallel (akin A000084); A051045 requires circuits to be combined with a single resistor at a time.
This sequence regards circuits as distinct only if their resistance is different; A006351 regards circuits distinct if their configuration is different, although some may have the same resistance.
This sequence considers resistors with contiguous resistances 1, 2, ..., n; A005840 considers arbitrarily different resistors, while A048211 considers n equal resistances.

			a(2) = 2 since given a 1-ohm and a 2-ohm resistor, a series circuit yields 3 ohms, while a parallel circuit yields 2/3 ohms, which thus yields two distinct resistances.

Cf. A005840, A006352, A048211, A051045.

cp's OEIS Frontend

A379460 Array read by downward antidiagonals: A(n,k) = A(n,k-1) + (k+1)*(A(n-1,k) + A(n-1,k+1)) with A(n,0) = A(n-1,0) + A(n-1,1), A(0,k) = 1, n >= 0, k >= 0.

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A298673 Inverse matrix of A135494.

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A379459 Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + 2(k+1)!Sum_{j=0..k} A(n-1,j)/j! with A(0,k) = 2*(k+1)!, n >= 0, k >= 0.

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A232005 Number of distinct resistances that can be produced from a circuit of resistors with resistances 1, 2, ..., n using only series and parallel combinations.

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cp's OEIS Frontend

A379460 Array read by downward antidiagonals: A(n,k) = A(n,k-1) + (k+1)*(A(n-1,k) + A(n-1,k+1)) with A(n,0) = A(n-1,0) + A(n-1,1), A(0,k) = 1, n >= 0, k >= 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A298673 Inverse matrix of A135494.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A379459 Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + 2*(k+1)!*Sum_{j=0..k} A(n-1,j)/j! with A(0,k) = 2*(k+1)!, n >= 0, k >= 0.

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Keywords

Examples

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Programs

PARI

Formula

A232005 Number of distinct resistances that can be produced from a circuit of resistors with resistances 1, 2, ..., n using only series and parallel combinations.

Views

Author

Keywords

Comments

Examples

Crossrefs

A379459 Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + 2(k+1)!Sum_{j=0..k} A(n-1,j)/j! with A(0,k) = 2*(k+1)!, n >= 0, k >= 0.