A249163 -id:A249163 - cp's OEIS frontend

A245602 Triangle read by rows: the negative terms of A163626.

-1, -3, -7, -6, -15, -60, -31, -390, -120, -63, -2100, -2520, -127, -10206, -31920, -5040, -255, -46620, -317520, -181440, -511, -204630, -2739240, -3780000, -362880, -1023, -874500, -21538440, -59875200, -19958400, -2047

Offset: 0

Paul Curtz, Dec 17 2014

These numbers a(n) are the companion of A249163(n).
Consider the Worpitzky fractions A163626(n)/A002260(n) yielding the second Bernoulli numbers A164555(n)/A027642(n):
1,
1, -1/2,
1, -3/2,  +2/3,
1, -7/2, +12/3, -6/4,
etc.
From the second row on, the sum of the numerators is 0.
The absolute values of every row of the numerators triangle A163626 are 1, 2, 6, 26, ... = A000629(n).
a(n) triangle is shifted. It starts from second row and second column of triangle above.
-1,
-3,
-7,       -6,
-15,     -60,
-31,    -390,    -120,
-63,   -2100,   -2520,
-127, -10206,  -31920,   -5040,
-255, -46620, -317520, -181440,
etc.
Sum of successive rows: -1, -3, -13, -75, ... = -A000670(n+1).
Successive columns: A000225, A028244, from the Stirling numbers of second kind S(n,2), S(n,4), S(n,6), S(n,8), S(n,10), ... . See A000770, A032180, A049434, A228910, A049435, A228912, A008277.
Thanks to Jean-François Alcover.

Cf. A000225, A000629, A000670, A000770, A002260, A008277, A009445, A027642, A028244, A032180, A049434, A049435, A163626, A164555, A228910, A228912, A249163.

Select[ Table[ (-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten, Negative] (* Jean-François Alcover, Dec 26 2014 *)

A258369 Stirling-Bernoulli transform of A027656.

Original entry on oeis.org

1, 1, 5, 25, 173, 1441, 14165, 160105, 2044733, 29105521, 456781925, 7834208185, 145760370893, 2923764916801, 62891469229685, 1444055265984265, 35250519098274653, 911569049328779281, 24893164161460525445, 715822742720760256345, 21620050147748210572013

Offset: 0

Philippe Deléham, May 28 2015

Also called Akiyama-Tanigawa transform of A027656.

			a(0) = 1*1 = 1.
a(1) = 1*1 = 1.
a(2) = 1*1 + 2*2 = 5.
a(3) = 1*1 + 12*2 = 25.
a(4) = 1*1 + 50*2 + 24*3 = 173.

Cf. A027656, A163626, A249163.

a(n) = Sum_{k = 0..n} A163626(n,k)*A027656(k).
a(n) = Sum_{k>=0} A249163(n,k) * (k+1).
E.g.f.: 1/(exp(x)*(2 - exp(x))^2).
a(n) ~ n! * n / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018

A249182 Proceed counterclockwise on the outer keys of a numeric keypad (i.e., 1,2,3,6,9,8,7,4): first single digits, then concatenate two digits, then three, etc.

Original entry on oeis.org

1, 2, 3, 6, 9, 8, 7, 4, 12, 23, 36, 69, 98, 87, 74, 41, 123, 236, 369, 698, 987, 874, 741, 412, 1236, 2369, 3698, 6987, 9874, 8741, 7412, 4123, 12369, 23698, 36987, 69874, 98741, 87412, 74123, 41236, 123698, 236987, 369874, 698741, 987412, 874123, 741236, 412369

Offset: 1

M. F. Hasler, Oct 23 2014

k-digit terms start at index 8k-7; for example, at index 57, start the 8-digit terms 12369874, 23698741, 36987412, 69874123, 98741236, 87412369, 74123698, 41236987.

Cf. A059805, A091331, A091332, A249163.

d=Vec("12369874");for(L=1,6,for(i=1,#d,print1(concat(vector(L,j,d[(i+j-2)%#d+1]))",")))

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A245602 Triangle read by rows: the negative terms of A163626.

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A258369 Stirling-Bernoulli transform of A027656.

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A249182 Proceed counterclockwise on the outer keys of a numeric keypad (i.e., 1,2,3,6,9,8,7,4): first single digits, then concatenate two digits, then three, etc.

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