A049398
a(n) = (n+9)!/9!.
Original entry on oeis.org
1, 10, 110, 1320, 17160, 240240, 3603600, 57657600, 980179200, 17643225600, 335221286400, 6704425728000, 140792940288000, 3097444686336000, 71241227785728000, 1709789466857472000, 42744736671436800000, 1111363153457356800000, 30006805143348633600000
Offset: 0
-
a049398 = (flip div 362880) . a000142 . (+ 9)
-- Reinhard Zumkeller, Aug 31 2014
-
[Factorial(n+9)/362880: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011
-
a[n_] := (n + 9)!/9!; Array[a, 20, 0] (* Amiram Eldar, Jan 15 2023 *)
-
a(n) = (n+9)!/9!
A051562
Second unsigned column of triangle A051380.
Original entry on oeis.org
0, 1, 19, 299, 4578, 71394, 1153956, 19471500, 343976400, 6366517200, 123418922400, 2503748556000, 53091962697600, 1175271048201600, 27123099523027200, 651708291206649600, 16282170039031142400
Offset: 0
- Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051380.
Cf.
A049389 (first unsigned column).
-
f[k_] := k + 8; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}]
(* Clark Kimberling, Dec 29 2011 *)
A051564
Second unsigned column of triangle A051523.
Original entry on oeis.org
0, 1, 21, 362, 6026, 101524, 1763100, 31813200, 598482000, 11752855200, 240947474400, 5154170774400, 114942011990400, 2669517204076800, 64496340380102400, 1619153396908185600, 42188624389562112000
Offset: 0
- Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051523.
Cf.
A049398 (first unsigned column).
-
f[n_] := n!*Sum[(-1)^k*Binomial[-10, k]/(n - k), {k, 0, n - 1}]; Array[f, 17, 0]
Range[0, 16]! CoefficientList[ Series[-Log[(1 - x)]/(1 - x)^10, {x, 0, 16}], x]
(* Or, using elementary symmetric functions: *)
f[k_] := k + 9; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}]
(* Clark Kimberling, Dec 29 2011 *)
A163936
Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).
Original entry on oeis.org
1, 1, 0, 2, 1, 0, 6, 8, 1, 0, 24, 58, 22, 1, 0, 120, 444, 328, 52, 1, 0, 720, 3708, 4400, 1452, 114, 1, 0, 5040, 33984, 58140, 32120, 5610, 240, 1, 0, 40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0, 362880, 3733920, 11026296, 12440064, 5765500, 1062500
Offset: 1
Triangle starts:
[ 1] 1;
[ 2] 1, 0;
[ 3] 2, 1, 0;
[ 4] 6, 8, 1, 0;
[ 5] 24, 58, 22, 1, 0;
[ 6] 120, 444, 328, 52, 1, 0;
[ 7] 720, 3708, 4400, 1452, 114, 1, 0;
[ 8] 5040, 33984, 58140, 32120, 5610, 240, 1, 0;
[ 9] 40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0;
The first few W1(z,p) polynomials are
W1(z,p=1) = 1/(1-z);
W1(z,p=2) = (1 + 0*z)/(1-z)^3;
W1(z,p=3) = (2 + 1*z + 0*z^2)/(1-z)^5;
W1(z,p=4) = (6 + 8*z + 1*z^2 + 0*z^3)/(1-z)^7.
-
with(combinat): a := proc(n, m): add((-1)^(n+k+1)*binomial(2*n-1, k)*stirling1(m+n-k-1, m-k), k=0..m-1) end: seq(seq(a(n, m), m=1..n), n=1..9); # Johannes W. Meijer, revised Nov 27 2012
-
Table[Sum[(-1)^(n + k + 1)*Binomial[2*n - 1, k]*StirlingS1[m + n - k - 1, m - k], {k, 0, m - 1}], {n, 1, 10}, {m, 1, n}] // Flatten (* G. C. Greubel, Aug 13 2017 *)
-
for(n=1,10, for(m=1,n, print1(sum(k=0,m-1,(-1)^(n+k+1)* binomial(2*n-1,k)*stirling(m+n-k-1,m-k, 1)), ", "))) \\ G. C. Greubel, Aug 13 2017
-
\\ assuming offset = 0:
E2poly(n,x) = if(n == 0, 1, x*(x-1)^(2*n)*deriv((1-x)^(1-2*n)*E2poly(n-1,x)));
{ for(n = 0, 9, print(Vec(E2poly(n,x)))) } \\ Peter Luschny, Feb 12 2021
A002775
a(n) = n^2 * n!.
Original entry on oeis.org
0, 1, 8, 54, 384, 3000, 25920, 246960, 2580480, 29393280, 362880000, 4829932800, 68976230400, 1052366515200, 17086945075200, 294226732800000, 5356234211328000, 102793666719744000, 2074369080655872000, 43913881247588352000, 973160803270656000000, 22531105497723863040000
Offset: 0
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Delbert L. Johnson, Table of n, a(n) for n = 0..447
- J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122.
- J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]
-
with(combinat):for n from 0 to 15 do printf(`%d, `,n!/2*sum(2*n, k=1..n)) od: # Zerinvary Lajos, Mar 13 2007
seq(sum(sum(mul(k, k=1..n),l=1..n),m=1..n), n=0..21); # Zerinvary Lajos, Jan 26 2008
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(1):seq(count(ZLL, size=n)*n^2, n=0..21); # Zerinvary Lajos, Jun 11 2008
a:=n->add(0+add(n!, j=1..n),j=1..n):seq(a(n), n=0..21); # Zerinvary Lajos, Aug 27 2008
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nn=20;a=1/(1-x); Range[0,nn]! CoefficientList[Series[x D[x D[a,x], x], {x,0,nn}], x] (* Geoffrey Critzer, Jan 17 2012 *)
Table[n^2 n!,{n,0,40}] (* Harvey P. Dale, Aug 01 2021 *)
A001233
Unsigned Stirling numbers of first kind s(n,6).
Original entry on oeis.org
1, 21, 322, 4536, 63273, 902055, 13339535, 206070150, 3336118786, 56663366760, 1009672107080, 18861567058880, 369012649234384, 7551527592063024, 161429736530118960, 3599979517947607200, 83637381699544802976, 2021687376910682741568, 50779532534302850198976, 1323714091579185857760000
Offset: 6
(-log(1-x))^6 = x^6 + 3*x^7 + 23/4*x^8 + 9*x^9 + ...
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
- F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
Drop[Abs[StirlingS1[Range[30],6]],5] (* Harvey P. Dale, Sep 17 2013 *)
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for(n=5,50,print1(polcoeff(prod(i=1,n,x+i),5,x),","))
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[stirling_number1(i,6) for i in range(6,22)] # Zerinvary Lajos, Jun 27 2008
A051431
a(n) = (n+10)!/10!.
Original entry on oeis.org
1, 11, 132, 1716, 24024, 360360, 5765760, 98017920, 1764322560, 33522128640, 670442572800, 14079294028800, 309744468633600, 7124122778572800, 170978946685747200, 4274473667143680000, 111136315345735680000, 3000680514334863360000, 84019054401376174080000
Offset: 0
A163940
Triangle related to the divergent series 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... for m >= -1.
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 1, 5, 3, 0, 1, 9, 17, 4, 0, 1, 14, 52, 49, 5, 0, 1, 20, 121, 246, 129, 6, 0, 1, 27, 240, 834, 1039, 321, 7, 0, 1, 35, 428, 2250, 5037, 4083, 769, 8, 0, 1, 44, 707, 5214, 18201, 27918, 15274, 1793, 9, 0, 1, 54, 1102, 10829, 54111, 133530, 145777, 55152, 4097, 10, 0
Offset: 0
The first few triangle rows are:
[1]
[1, 0]
[1, 2, 0]
[1, 5, 3, 0]
[1, 9, 17, 4, 0]
[1, 14, 52, 49, 5, 0]
The first few M(x,m) are:
M(x,m=0) = 1
M(x,m=1) = 1 + 0*x
M(x,m=2) = 1 + 2*x + 0*x^2
M(x,m=3) = 1 + 5*x + 3*x^2 + 0*x^3
The first few ST(x,m) are:
ST(x,m=-1) = 1
ST(x,m=0) = 1 + 0*x
ST(x,m=1) = 1 + 1*x + 0*x^2
ST(x,m=2) = 1 + 3*x + x^2 + 0*x^3
ST(x,m=3) = 1 + 6*x + 7*x^2 + x^3 + 0*x^4
The first few g(x,m) are:
g(x,-1) = (-1)*(- (1)*Ei(1,1/x)*exp(1/x))/x^0
g(x,0) = (1)*((1)*x - (1)*Ei(1,1/x)*exp(1/x))/x^1
g(x,1) = (-1)*((1)*x - (1+ x)*Ei(1,1/x)*exp(1/x))/x^2
g(x,2) = (1)*((1+2*x)*x - (1+3*x+x^2)*Ei(1,1/x)*exp(1/x))/x^3
g(x,3) = (-1)*((1+5*x+3*x^2)*x - (1+6*x+7*x^2+x^3)*Ei(1,1/x)*exp(1/x))/x^4
The row sums equal
A040027 (Gould).
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nmax := 10; for p from 1 to nmax do Gf(p) := convert(series(1/((1-(p-1)*x)^2*product((1-k1*x), k1=1..p-2)), x, nmax+1-p), polynom); for q from 0 to nmax-p do a(p+q-1, q) := coeff(Gf(p), x, q) od: od: seq(seq(a(n, k), k=0..n), n=0..nmax-1);
# End program 1
nmax1:=nmax; A040027 := proc(n): if n = -1 then 0 elif n= 0 then 1 else add(binomial(n, k1-1)*A040027(n-k1), k1 = 1..n) fi: end: A000110 := proc(n) option remember; if n <= 1 then 1 else add( binomial(n-1, i) * A000110(n-1-i), i=0..n-1); fi; end: A073003 := - exp(1) * Ei(-1): for n from -1 to nmax1 do g(1, n) := (-1)^n * (A040027(n) - A000110(n+1) * A073003) od;
# End program 2
-
nmax = 11;
For[p = 1, p <= nmax, p++, gf = 1/((1-(p-1)*x)^2*Product[(1-k1*x), {k1, 1, p-2}]) + O[x]^(nmax-p+1) // Normal; For[q = 0, q <= nmax-p, q++, a[p+q-1, q] = Coefficient[gf, x, q]]];
Table[a[n, k], {n, 0, nmax-1}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 02 2019, from 1st Maple program *)
A165674
Triangle generated by the asymptotic expansions of the E(x,m=2,n).
Original entry on oeis.org
1, 3, 1, 11, 5, 1, 50, 26, 7, 1, 274, 154, 47, 9, 1, 1764, 1044, 342, 74, 11, 1, 13068, 8028, 2754, 638, 107, 13, 1, 109584, 69264, 24552, 5944, 1066, 146, 15, 1, 1026576, 663696, 241128, 60216, 11274, 1650, 191, 17, 1
Offset: 1
A093905 is the reversal of this triangle.
A080663, n>=2, is the third right hand column.
A094587 is generated by the asymptotic expansions of E(x, m=1, n).
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nmax:=9; for n from 1 to nmax do a(n, n) := 1 od: for n from 2 to nmax do a(n, 1) := n*a(n-1, 1) + (n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do a(n, m) := (n-m+1)*a(n-1, m) + a(n-1, m-1) od: od: seq(seq(a(n, m), m = 1..n), n = 1..nmax);
# End program 1
nmax := nmax+1: m:=2; with(combinat): EA := proc(x, m, n) local E, i; E:=0: for i from m-1 to nmax+2 do E := E + sum((-1)^(m+k1+1) * binomial(k1, m-1) * n^(k1-m+1) * stirling1(i, k1), k1=m-1..i) / x^(i-m+1) od: E:= exp(-x)/x^(m) * E: return(E); end: for n1 from 1 to nmax do f(n1-1) := simplify(exp(x) * x^(nmax+3) * EA(x, m, n1)); for m1 from 0 to nmax+2 do b(n1-1, m1) := coeff(f(n1-1), x, nmax+2-m1) od: od: for n1 from 0 to nmax-1 do for m1 from 0 to n1-m+1 do a(n1-m+2, m1+1) := abs(b(m1, n1-m1)) od: od: seq(seq(a(n, m), m = 1..n),n = 1..nmax-1);
# End program 2
# Maple programs revised by Johannes W. Meijer, Sep 22 2012
A001234
Unsigned Stirling numbers of the first kind s(n,7).
Original entry on oeis.org
1, 28, 546, 9450, 157773, 2637558, 44990231, 790943153, 14409322928, 272803210680, 5374523477960, 110228466184200, 2353125040549984, 52260903362512720, 1206647803780373360, 28939583397335447760
Offset: 7
G.f. = x^7 + 28*x^8 + 546*x^9 + 9450*x^10 + 157773*x^11 + 2637558*x^12 + ...
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 834.
- F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
A001234 := proc(n) abs(combinat[stirling1](n,7)) ; end: seq(A001234(n),n=7..30) ; # R. J. Mathar, Nov 06 2009
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Table[Abs[StirlingS1[n, 7]], {n, 7, 40}] (* Jean-François Alcover, Mar 24 2020 *)
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for(n=6,50,print1(polcoeff(prod(i=1,n,x+i),6,x),","))
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[stirling_number1(i,7) for i in range(7,22)] # Zerinvary Lajos, Jun 27 2008
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