A094638 -id:A094638 - cp's OEIS frontend

A100492 Triangle read by rows giving the coefficients of general sum formulas of n-th Fibonacci numbers (A000045). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies F(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

1, -1, -4, -3, 10, 49, 95, 83, 27, -90, -740, -2415, -4110, -3890, -1950, -405, 1320, 14054, 64116, 164059, 258461, 257604, 159070, 55755, 8505, -23640, -318684, -1881532, -6452300, -14294605, -21442540, -22106669, -15496012, -7078575, -1905120, -229635, 523440, 8474100, 61424596

Offset: 1

André F. Labossière, Nov 22 2004

			F(7) = (1/(7-1)!) * [ 7^(7-1) -{1+4*(7-2)+3*C(7-2,2)}*7^(7-2) +{10+49*(7-3)+95*C(7-3,2)+83*C(7-3,3) +27*C(7-3,4)}*7^(7-3) -{90+740*(7-4)+2415*C(7-4,2)+4110*C(7-4,3)}*7^(7-4) +... ]
= (1/6!) * [ 7^6 -{1+20+30}*7^5 +{10+196+570+332+27}*7^4 -{90+2220+7245+4110}*7^3 +{1320+28108 +64116}*7^2 -{23640+318684}*7 +{523440} ]
= (1/6!) * [ 7^6 -51*7^5 +1135*7^4 -13665*7^3 +93544*7^2 -342324*7 +523440 ]
= (1/720) * [ 117649 -857157 +2725135 -4687095 +4583656 -2396268 +523440 ] = 9360/720 = 13.

Cf. A099731, A000045, A094216, A094638, A000108.

A101559 This table (read by rows) shows the coefficients of sum formulas of n-th subfactorial numbers (A000166). The n-th row (n>=1) contains T(i,n) for i=1 to n, where T(i,n) satisfies Subf(n) = Sum_{i=1..n} T(i,n) * n^(n-i).

Original entry on oeis.org

1, 1, -2, 1, -4, 4, 1, -7, 15, -10, 1, -11, 42, -65, 34, 1, -16, 96, -267, 339, -154, 1, -22, 191, -831, 1891, -2103, 874, 1, -29, 344, -2151, 7600, -15023, 15171, -5914, 1, -37, 575, -4880, 24600, -74884, 133147, -124755, 46234, 1, -46, 907, -10025, 68153, -293925, 798564, -1305847, 1151331, -409114, 1, -56

Offset: 1

André F. Labossière, Dec 06 2004

			Subf(9) = [ 9^8 -37*9^7 +575*9^6 -4880*9^5 +24600*9^4 -74884*9^3 +133147*9^2 - 124755*9 +46234 ] = 14833.

Cf. A101560, A000166, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.

A124380 O.g.f.: A(x) = Sum_{n>=0} x^nProduct_{k=0..n} (1 + kx).

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 57, 157, 453, 1368, 4296, 13995, 47138, 163779, 585741, 2152349, 8113188, 31326760, 123748871, 499539900, 2058542819, 8651755865, 37054078481, 161591063250, 717032333816, 3235298221401, 14834735654080, 69085973044125

Offset: 0

Paul D. Hanna, Oct 28 2006

nonn

The Kn11 triangle sums of A094638 are given by the terms of this sequence. For the definitions of this and other triangle sums see A180662. [Johannes W. Meijer, Apr 20 2011]

			A(x) = 1 + x*(1+x) + x^2*(1+x)*(1+2x) + x^3*(1+x)*(1+2x)*(1+3x) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..868
Giulio Cerbai, Anders Claesson, and Bruce E. Sagan, Self-modified difference ascent sequences, arXiv:2408.06959 [math.CO], 2024. See p. 15.

Cf. A024427, A171367.

nmax = 30; CoefficientList[Series[Sum[x^(2*k)*Pochhammer[1 + 1/x, k], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 14 2024 *)
Table[Sum[(-1)^k * StirlingS1[n+1-k, n+1-2*k], {k, 0, (n+1)/2}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 18 2024 *)

a(n)=polcoeff(sum(k=0,n,x^k*prod(j=0,k,1+j*x+x*O(x^n))),n)

O.g.f.: A(x) = 1 + x*(1+x)/(G(0) - x*(1+x)) ; G(k) = 1+x*(k*x+x+1) - x*(k*x + 2*x + 1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 02 2011
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - (1+x*k)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: 1/(x*Q(0)-1)/x^4 + (1+x-x^3)/x^4, where Q(k)= 1 - x/(1 - (k+1)*x - x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
Conjecture: log(a(n)) ~ n*log(n)/2 - n*(1 + log(2))/2. - Vaclav Kotesovec, Sep 18 2024

A004130 Numerators in expansion of (1-x)^{-1/4}.

Original entry on oeis.org

1, 1, 5, 15, 195, 663, 4641, 16575, 480675, 1762475, 13042315, 48612265, 729183975, 2748462675, 20809788825, 79077197535, 4823709049635, 18443593425075, 141400882925575, 543277076503525, 8366466978154285, 32270658344309385

Offset: 0

N. J. A. Sloane

Numerators in expansion of sqrt(1/sqrt(1-4x)). - Paul Barry, Jul 12 2005

Denominators are in A088802. - Michael Somos, Aug 23 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A004134, A004981, A034255, A034385, A048882, A007696, A000265, A049606.

Table[Numerator[Binomial[-1/4, n] (-1)^n], {n, 0, 20}]

{a(n) = if( n<0, 0, numerator( polcoeff( (1 - x +x*O(x^n))^(-1/4), n ) ) ) } /* Michael Somos, Aug 23 2007 */

a(n) = prod(k=1, n, (4k-3)/k * 2^A007814(k)), proved by Mitch Harris, following a conjecture by Ralf Stephan.
a(n) = 2^(e_2((2n)!)-n)/n! Product[4k+1,{k,0,n-1}], where e_2((2n)!) is the highest power of 2 that divides (2n)! (sequence A005187). - Emanuele Munarini, Jan 25 2011
Numerators in (1-4t)^(-1/4) = 1 + t + (5/2)t^2 + (15/2)t^3 + (195/8)t^4 + (663/8)t^5 + (4641/16)t^6 + (16575/16)t^7 + ... = 1 + t + 5*t^2/2! + 45*t^3/3! + 585*t^4/4! + ... = e.g.f. for the quartic factorials A007696 (cf. A094638). - Tom Copeland, Dec 04 2013

A055505 Numerators in expansion of (1-x)^(-1/x)/e.

Original entry on oeis.org

1, 1, 11, 7, 2447, 959, 238043, 67223, 559440199, 123377159, 29128857391, 5267725147, 9447595434410813, 1447646915836493, 225037938358318573, 29911565062525361, 3651003047854884043877, 38950782815463986767

Offset: 0

N. J. A. Sloane, Jul 11 2000

From Miklos Kristof, Nov 04 2007: (Start) This is also the sequence of numerators associated with expansion of (1+x)^(1/x).
(1 + x)^(1/x) = exp(1)*(1 - 1/2*x + 11/24*x^2 - 7/16*x^3 + 2447/5760*x^4 - 959/2304*x^5 + 238043/580608*x^6 - ...).
(1+x)^(1/x) = exp(log(1+x)/x) = exp(1)*exp(-x/2)*exp(x^2/3)*exp(x^3/4)*...
Let a(n) be this sequence, let b(n) be A055535. Then (1+x)^(1/x)=exp(1)*a(n)/b(n) x^n.
a(n)/b(n) = Sum_{i>=n} s(i,i-n)/i! where s(n,m) is a Stirling number of the first kind.
exp(1) = 1 + Sum_{i>=1} s(i,i)/i!, for the n=1 case.
a(1)/b(1) = 1/1 because 1+1/1!+1/2!+1/3!+1/4!+... = exp(1)
a(2)/b(2) = 1/2 because 1/2!+3/3!+6/4!+10/5!+... = 1/2*exp(1)
a(3)/b(3) = 11/24 because 2/3!+11/4!+35/5!+85/6!+... = 11/24*exp(1)
a(4)/b(4) = 7/16 because 6/4!+50/5!+225/6!+735/7!+... = 7/16*exp(1) (End)

			1+1/2*x+11/24*x^2+7/16*x^3+2447/5760*x^4+...
1, -1/2, 11/24, -7/16, 2447/5760, -959/2304, 238043/580608, -67223/165888, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.1.

G. C. Greubel, Table of n, a(n) for n = 0..250
Markus Brede, On the convergence of the sequence defining Euler's number, Math. Intelligencer, 27, no. 3 (2005), 6-7.
Chao-Ping Chen and Junesang Choi, An Asymptotic Formula for (1+1/x)^x Based on the Partition Function, Amer. Math. Monthly 121 (2014), no. 4, 338--343. MR3183017.
Branko Malesevic, Yue Hu, and Cristinel Mortici, Accurate Estimates of (1+x)^{1/x} Involved in Carleman Inequality and Keller Limit, arXiv:1801.04963 [math.CA], 2018.

Cf. A094638, A130534, A055535 (denominators).
See also A239897/A239898.
Cf. A276977.

T:= proc(u) local k, l; add( Stirling1(u+k,k)*((u+k)!)^(-1)* add( (-1)^l/l!, l=0..u-k), k=0..u); end;

a[n_] := Sum[StirlingS1[n+k, k]/(n+k)!*Sum[(-1)^j/j!, {j, 0, n-k}], {k, 0, n}]; Table[a[n] // Numerator // Abs, {n, 0, 17}] (* Jean-François Alcover, Mar 04 2014, after Maple *)
Numerator[((1-x)^(-1/x)/E + O[x]^20)[[3]]] (* or *)
Numerator[Table[Sum[StirlingS1[n+k, k] Subfactorial[n-k] Binomial[2n, n+k], {k, 0, n}] (-1)^n/(2n)!, {n, 0, 10}]] (* Vladimir Reshetnikov, Sep 23 2016 *)

See Maple line for formula.

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

A055535 Denominators in expansion of (1-x)^(-1/x)/e.

Original entry on oeis.org

1, 2, 24, 16, 5760, 2304, 580608, 165888, 1393459200, 309657600, 73574645760, 13377208320, 24103053950976000, 3708162146304000, 578473294823424000, 77129772643123200, 9440684171518279680000, 100969884187361280000

Offset: 0

N. J. A. Sloane, Jul 11 2000

Or, equally, denominators in expansion of (1+x)^(1/x)/e.

			(1-x)^(-1/x) = exp(1)*(1 + 1/2*x + 11/24*x^2 + 7/16*x^3 + 2447/5760*x^4 + 959/2304*x^5 + 238043/580608*x^6 + ...).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.1.

Robert Israel, Table of n, a(n) for n = 0..377
Chao-Ping Chen and Junesang Choi, An Asymptotic Formula for (1+1/x)^x Based on the Partition Function, Amer. Math. Monthly 121 (2014), no. 4, 338--343. MR3183017.

Cf. A094638, A130534, A055505 (numerators), A276977.

G:= (1-x)^(-1/x)/exp(1):
S:= series(G,x,32):
seq(denom(coeff(S,x,j)),j=0..30); # Robert Israel, Sep 23 2016

a[n_] := Sum[StirlingS1[n+k, k]/(n+k)!*Sum[(-1)^j/j!, {j, 0, n-k}], {k, 0, n}]; Table[a[n] // Denominator, {n, 0, 17}] (* Jean-François Alcover, Mar 04 2014 *)
Denominator[((1+x)^(1/x)/E + O[x]^20)[[3]]] (* or *)
Denominator[Table[Sum[StirlingS1[n+k, k] Subfactorial[n-k] Binomial[2n, n+k], {k, 0, n}]/(2n)!, {n, 0, 10}]] (* Vladimir Reshetnikov, Sep 23 2016 *)

From Miklos Kristof, Nov 04 2007 (Start):
(1+x)^(1/x) = exp(log(1+x)/x) = exp(1)*exp(-x/2)*exp(x^2/3)*exp(x^3/4)*...
Let a(n) be A055505, let b(n) be this sequence. Then (1+x)^(1/x) = exp(1)*a(n)/b(n) x^n.
a(n)/b(n) = Sum_{i>=n} s(i,i-n)/i! where s(n,m) is a Stirling number of the first kind.
exp(1) = 1 + Sum_{i>=1} s(i,i)/i! for the n = 1 case.
a(1)/b(1) = 1/1 because 1+1/1!+1/2!+1/3!+1/4!+... = exp(1)
a(2)/b(2) = 1/2 because 1/2!+3/3!+6/4!+10/5!+... = 1/2*exp(1)
a(3)/b(3) = 11/24 because 2/3!+11/4!+35/5!+85/6!+... = 11/24*exp(1)
a(4)/b(4) = 7/16 because 6/4!+50/5!+225/6!+735/7!+... = 7/16*exp(1) (End)

Edited by N. J. A. Sloane, Jul 25 2008 at the suggestion of R. J. Mathar and Eric Rowland

A232773 Permanent of the n X n matrix with numbers 1,2,...,n^2 in order across rows.

Original entry on oeis.org

1, 1, 10, 450, 55456, 14480700, 6878394720, 5373548250000, 6427291156586496, 11157501095973529920, 26968983444160450560000, 87808164603589940623344000, 374818412822626584819196231680, 2050842983500342507649178541536000, 14112022767608502582976078751055052800

Offset: 0

Franklin T. Adams-Watters, Nov 30 2013

nonn

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A114533, A232788, A008277, A232818, A204248, A094638.

a:= n-> (-1)^n*add(n^k*Stirling1(n, n-k)*
        Stirling1(n+1, k+1)*(n-k)!*k!, k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 02 2013

Table[(-1)^n * Sum[n^k * StirlingS1[n, n-k] * StirlingS1[n+1, k+1] * (n-k)! * k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec after Max Alekseyev, Nov 30 2013 *)

a(n) = (-1)^n * sum(k=0,n, n^k * stirling(n,n-k) * stirling(n+1,k+1) * (n-k)! * k! ) /* Max Alekseyev, Nov 30 2013 */

from sympy.functions.combinatorial.numbers import stirling, factorial
def A232773(n): return abs(sum(n**k*stirling(n,n-k,kind=1,signed=True)*stirling(n+1,k+1,kind=1,signed=True)*factorial(n-k)*factorial(k) for k in range(n+1))) # Chai Wah Wu, Mar 25 2025

a(n) = (-1)^n * Sum_{k=0..n} n^k * Stirling1(n,n-k) * Stirling1(n+1,k+1) * (n-k)! * k!. - Max Alekseyev, Nov 30 2013
Limit_{n->oo} a(n)^(1/n)/n^3 = exp(-2). - Vaclav Kotesovec, Nov 30 2013
a(n) = A232788(n)*n!!, where n!! = A006882(n) is the double-factorial. - M. F. Hasler, Nov 30 2013

More terms from W. Edwin Clark, Nov 30 2013

a(0)=1 prepended by Alois P. Heinz, Dec 02 2013

A101033 Triangle read by rows giving the coefficients of general sum formulas of n-th Lucas numbers (A000204). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies L(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

Original entry on oeis.org

1, 1, -2, -3, 2, 15, 51, 65, 27, 6, -148, -945, -2292, -2776, -1680, -405, 24, 2290, 19580, 71965, 145525, 175244, 125950, 50085, 8505, 120, -41676, -473072, -2340400, -6676835, -12132890, -14587261, -11619692, -5290005, -1752030, -229635, 720, 943908, 13132532, 81977672, 303352938, 740797855

Offset: 1

André F. Labossière, Nov 30 2004

			L(7)= (1/(7-1)!) * [ 7^(7-1) -{-1+2*(7-2)+3*C(7-2,2)}*7^(7-2) +{2+15*(7-3)+51*C(7-3,2)+65*C(7-3,3) +27*C(7-3,4)}*7^(7-3) -{-6+148*(7-4)+945*C(7-4,2)+2292*C(7-4,3)}*7^(7-4) +... ]
= (1/6!) * [ 7^6 -{-1+10+30}*7^5 +{2+60+306+260+27}*7^4 -{-6+444+2835+2292}*7^3 +{24+4580+19580}*7^2 -{-120+41676}*7 +{720} ] = (1/6!) * [ 7^6 -39*7^5 +655*7^4 -5565*7^3 +24184*7^2 -41556*7 +720 ]
= (1/720) * [ 117649 -655473 +1572655 -1908795 +1185016 -290892 +720 ] = 20880/720 = 29.

Ron Knott, The Lucas Numbers in Pascal's Triangle.

Cf. A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.

A102411 Even triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies !n = Sum_{i=1..k+2} T(i,k) n^(i-1) / (2*k-2)!.

Original entry on oeis.org

0, 1, 0, -16, 5, 1, 0, 5256, -3068, 276, 32, 0, 2070720, 2367420, -912150, 53220, 3510, 0, -36031524480, 15327895296, -40587120, -387492840, 21414120, 758184, 840, -212459319878400, -75473246681280, 38182549456800, -2562251680800, -195611371200, 13639812480, 285616800, 453600

Offset: 1

André F. Labossière, Jan 07 2005

The sum of signed coefficients for each k-th row is divisible by (2*k-2)!. Moreover, another variant (but an incomplete one, and sorted differently) of the above sequence is presented in A101752.

			Triangle starts:
0, 1, 0;
-16, 5, 1, 0;
5256, -3068, 276, 32, 0;
2070720, 2367420, -912150, 53220, 3510, 0;
-36031524480, 15327895296, -40587120, -387492840, 21414120, 758184, 840;
...
!11=4037914; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing !11.
=> !11 = [ -212459319878400 -75473246681280*11 +38182549456800*11^2 -2562251680800*11^3 -195611371200*11^4 +13639812480*11^5 +285616800*11^6 +453600*11^7 ]/10! = 4037914.

Cf. A102412, A094638, A094216, A003422, A008276, A101752, A102409, A102410, A101751, A000142, A101559, A101032, A099731.

A102412 Odd triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+1, where k=[2n+3+(-1)^n]/4 and T(i,k) satisfies !n = Sum_{i=1..k+1} T(i,k) n^(i-1) / (2*k-2)!.

Original entry on oeis.org

0, 1, -4, 4, 0, 96, -396, 108, 0, 1012320, -192900, -64890, 11460, 90, -2038014720, 1977810240, -304486560, -12131280, 2792160, 21840, -33190735737600, 4445760574080, 2334485260800, -394554283200, 2330344800, 1198048320, 8215200

Offset: 1

André F. Labossière, Jan 07 2005

Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!.

			Triangle starts:
0, 1;
-4, 4, 0;
96, -396, 108, 0;
1012320, -192900, -64890, 11460, 90;
-2038014720, 1977810240, -304486560, -12131280, 2792160, 21840;
...
!11=4037914; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing !11.
=> !11 = [ -33190735737600 +4445760574080*11 +2334485260800*11^2 -394554283200*11^3 +2330344800*11^4 +1198048320*11^5 +8215200*11^6 ]/10! = 4037914.

Cf. A102411, A094638, A094216, A003422, A008276, A101752, A102409, A102410, A101751, A000142, A101559, A101032, A099731.

cp's OEIS Frontend

A100492 Triangle read by rows giving the coefficients of general sum formulas of n-th Fibonacci numbers (A000045). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k-1, where T(i,k) satisfies F(n) = Sum_{k=1..n} Sum_{i=1..2*k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

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A101559 This table (read by rows) shows the coefficients of sum formulas of n-th subfactorial numbers (A000166). The n-th row (n>=1) contains T(i,n) for i=1 to n, where T(i,n) satisfies Subf(n) = Sum_{i=1..n} T(i,n) * n^(n-i).

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A124380 O.g.f.: A(x) = Sum_{n>=0} x^n*Product_{k=0..n} (1 + k*x).

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A004130 Numerators in expansion of (1-x)^{-1/4}.

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A055505 Numerators in expansion of (1-x)^(-1/x)/e.

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A055535 Denominators in expansion of (1-x)^(-1/x)/e.

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A232773 Permanent of the n X n matrix with numbers 1,2,...,n^2 in order across rows.

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A101033 Triangle read by rows giving the coefficients of general sum formulas of n-th Lucas numbers (A000204). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k-1, where T(i,k) satisfies L(n) = Sum_{k=1..n} Sum_{i=1..2*k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

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A100492 Triangle read by rows giving the coefficients of general sum formulas of n-th Fibonacci numbers (A000045). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies F(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

A124380 O.g.f.: A(x) = Sum_{n>=0} x^nProduct_{k=0..n} (1 + kx).

A101033 Triangle read by rows giving the coefficients of general sum formulas of n-th Lucas numbers (A000204). The k-th row (k>=1) contains T(i,k) for i=1 to 2k-1, where T(i,k) satisfies L(n) = Sum_{k=1..n} Sum_{i=1..2k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.

A102411 Even triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies !n = Sum_{i=1..k+2} T(i,k) n^(i-1) / (2*k-2)!.

A102412 Odd triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+1, where k=[2n+3+(-1)^n]/4 and T(i,k) satisfies !n = Sum_{i=1..k+1} T(i,k) n^(i-1) / (2*k-2)!.