A014145 -id:A014145 - cp's OEIS frontend

A370369 a(n) = n! + Sum_{k=1..n-1} (n-k)*k! = n! + A014145(n-1); for n >= 2, number of m such that any two consecutive digits of the base-n expansion of m differ by 1 after arranging the digits in decreasing order.

1, 3, 10, 37, 166, 919, 6112, 47305, 416098, 4091131, 44417044, 527456557, 6798432070, 94499679583, 1408924024696, 22425642181009, 379514672913322, 6804212771165635, 128827325000617948, 2568509718703606261, 53787877376348226574, 1180349932648067726887, 27086018941198865627200

Offset: 1

Jianing Song, Feb 16 2024

Given n, the largest such number is Sum_{i=0..n-1} i*n^i = A062813(n). If zero is excluded, the number of such k with d digits in base n, 1 <= d <= n, is (n+1-d)*d! - (d-1)!.

			a(3) = 10 because such numbers are 0_3, 1_3, 2_3, 10_3, 12_3, 21_3, 102_3, 120_3, 201_3 and 210_3.
a(10) = 4091131 is the number of terms of A215014.

Cf. A215014, A014145, A062813.

PARI
```
a(n) = n! + sum(k=1, n-1, (n-k)*k!)
```

A288777 Triangle read by rows in which column k lists the positive multiples of the factorial of k, with 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 6, 4, 6, 12, 24, 5, 8, 18, 48, 120, 6, 10, 24, 72, 240, 720, 7, 12, 30, 96, 360, 1440, 5040, 8, 14, 36, 120, 480, 2160, 10080, 40320, 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880, 10, 18, 48, 168, 720, 3600, 20160, 120960, 725760, 3628800, 11, 20, 54, 192, 840, 4320, 25200, 161280, 1088640

Offset: 1

Omar E. Pol, Jun 15 2017

T(n,k) is the number of k-digit numbers in base n+1 with distinct positive digits that form an integer interval when sorted.
T(9,k) is also the number of numbers with k digits in A288528.
The number of terms in A288528 is also A014145(9) = 462331, the same as the sum of the 9th row of this triangle.
Removing the left column of A137267 and of A137948 then this triangle appears in both cases.

			Triangle begins:
   1;
   2,  2;
   3,  4,  6;
   4,  6, 12,  24;
   5,  8, 18,  48, 120;
   6, 10, 24,  72, 240,  720;
   7, 12, 30,  96, 360, 1440,  5040;
   8, 14, 36, 120, 480, 2160, 10080,  40320;
   9, 16, 42, 144, 600, 2880, 15120,  80640,  362880;
  10, 18, 48, 168, 720, 3600, 20160, 120960,  725760, 3628800;
  11, 20, 54, 192, 840, 4320, 25200, 161280, 1088640, 7257600, 39916800;
  ...
For n = 9 and k = 2: T(9,2) is the number of numbers with two digits in A288528.
For n = 9 the row sum is 9 + 16 + 42 + 144 + 600 + 2880 + 15120 + 80640 + 362880 = 462331, the same as A014145(9) and also the same as the number of terms in A288528.

Right border gives A000142, n>=1.
Middle diagonal gives A001563, n>=1.
Row sums give A014145, n>=1.
Column 1..4: A000027, A005843, A008588, A008606.
Cf. A007489, A000142, A004736, A137267, A137948, A166350, A288528, A288778, A288780.

Table[(n - k + 1) k!, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Jun 15 2017 *)

T(n,k) = (n-k+1)*k! = (n-k+1)*A000142(k) = A004736(n,k)*A166350(n,k).
T(n,k) = Sum_{j=1..n} A166350(j,k).
T(n,k) =  A288778(n,k) + A000142(k-1).

A094344 Triangle T(n,k), 0<= k <= n, read by rows; given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] DELTA [1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 4, 1, 0, 6, 18, 13, 1, 0, 24, 96, 119, 46, 1, 0, 120, 600, 1059, 777, 199, 1, 0, 720, 4320, 9890, 10760, 5536, 1072, 1, 0, 5040, 35280, 99158, 142990, 111316, 44228, 6985, 1, 0, 40320, 322560, 1073692, 1926312, 2009578, 1217352, 395865, 53218, 1

Offset: 0

Philippe Deléham, Jun 02 2004

			Triangle begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  4,   1;
  0,  6, 18,  13,  1;
  0, 24, 96, 119, 46, 1;
  ...

Row sums: A094664.
Columns: A000007, A000142, A001563.
Diagonals: A000012, A014145.

Sum_{k=0..n} T(n,k)*3^(n-k) = A128709(n). - Philippe Deléham, Mar 27 2007

Row 9 completed by Michel Marcus, Jun 20 2023

A200545 Triangle T(n,k), read by rows, given by (1,0,2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 13, 9, 1, 0, 1, 46, 56, 16, 1, 0, 1, 199, 334, 160, 25, 1, 0, 1, 1072, 2157, 1408, 365, 36, 1, 0, 1, 6985, 15701, 12445, 4417, 721, 49, 1, 0, 1, 53218, 129214, 116698, 50944, 11452, 1288, 64, 1, 0, 1, 462331, 1191336, 1183216, 597026, 166716, 25956, 2136, 81, 1, 0

Offset: 0

Philippe Deléham, Nov 19 2011

Row sums : A000142(n) = n!.

			Triangle begins :
1
1, 0
1, 1, 0
1, 4, 1, 0
1, 13, 9, 1, 0
1, 46, 56, 16, 1, 0
1, 199, 334, 160, 25, 1, 0
1, 1072, 2157, 1408, 365, 36, 1, 0
1, 6985, 15701, 12445, 4417, 721, 49, 1, 0
1, 53218, 129214, 116698, 50944, 11452, 1288, 64, 1, 0

Alois P. Heinz, Rows n = 0..140, flattened
Sergey Kitaev, Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.

Cf. A000290, A014145,

DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x*r[[k + 1]] + y*s[[k + 1]]; p[0, ] = 1; p[, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k - 1] + q[k]*p[n - 1, k + 1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n - k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]];
m = 10;
DELTA[LinearRecurrence[{1, 1, -1}, {1, 0, 2}, m], LinearRecurrence[{0, 1}, {0, 1}, m], m] // Flatten (* Jean-François Alcover, Feb 21 2019 *)

Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A172485(n+1), A146559(n), A000012(n), A000142(n), A003319(n), A111529(n), A111530(n), A111531(n), A111532(n), A111533(n) for x = -2,-1,0,1,2,3,4,5,6,7 respectively.
T(k+2,k)=(k+1)^2 = A000290(k+1).
T(n+1,1)= A014145(n).

A288528 Numbers with consecutive positive decimal digits after the digits are sorted.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 123, 132, 213, 231, 234, 243, 312, 321, 324, 342, 345, 354, 423, 432, 435, 453, 456, 465, 534, 543, 546, 564, 567, 576, 645, 654, 657, 675, 678, 687, 756, 765, 768, 786, 789, 798, 867, 876, 879, 897, 978, 987

Offset: 1

Omar E. Pol, Jun 15 2017

The last term is a(462331) = 987654321.
Observation: the number of terms mentioned above is also A014145(9). Also the sum of the 9th row in the triangle A288777.
It appears that the number of terms with k digits in this sequence is also A288777(9,k), k>=1.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Subsequence of A215014.
Supersequence of A138141.
Cf. A014145, A033075, A288777.

def ok(n): return "".join(sorted(str(n))) in "123456789"
print([k for k in range(999) if ok(k)]) # Michael S. Branicky, Aug 04 2022

# alternate for generating full sequence instantly
from itertools import permutations
frags = ["123456789"[i:j] for i in range(9) for j in range(i+1, 10)]
afull = sorted(int("".join(s)) for f in frags for s in permutations(f))
print(afull[:70]) # Michael S. Branicky, Aug 04 2022

cp's OEIS Frontend

A370369 a(n) = n! + Sum_{k=1..n-1} (n-k)*k! = n! + A014145(n-1); for n >= 2, number of m such that any two consecutive digits of the base-n expansion of m differ by 1 after arranging the digits in decreasing order.

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A288777 Triangle read by rows in which column k lists the positive multiples of the factorial of k, with 1 <= k <= n.

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A094344 Triangle T(n,k), 0<= k <= n, read by rows; given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] DELTA [1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] where DELTA is the operator defined in A084938.

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Extensions

A200545 Triangle T(n,k), read by rows, given by (1,0,2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.

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Mathematica

Formula

A288528 Numbers with consecutive positive decimal digits after the digits are sorted.

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