A095000
E.g.f.: exp(x)/(1-x)^4.
Original entry on oeis.org
1, 5, 29, 193, 1457, 12341, 116125, 1203329, 13627073, 167525317, 2222710781, 31665408545, 482196718129, 7817359305653, 134443910166077, 2444991262876321, 46883166605035265, 945426638499719429, 20002372214708227933, 443036881445294292737, 10252840082607606694961
Offset: 0
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a := n -> hypergeom([4, -n], [], -1); seq(round(evalf(a(n), 100)), n=0..18); # Peter Luschny, Sep 20 2014
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Table[n!*SeriesCoefficient[E^(x)/(1-x)^4,{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
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x='x+O('x^66); Vec(serlaplace(exp(x)/(1-x)^4)) \\ Joerg Arndt, May 11 2013
A276588
Square array A(row,col) = Sum_{k=0..row} binomial(row,k)*(1+col+k)!, read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...
Original entry on oeis.org
1, 2, 3, 6, 8, 11, 24, 30, 38, 49, 120, 144, 174, 212, 261, 720, 840, 984, 1158, 1370, 1631, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 362880, 403200, 448560, 499680, 557400, 622704, 696750, 780908, 876809, 3628800, 3991680, 4394880, 4843440, 5343120, 5900520, 6523224, 7219974, 8000882, 8877691
Offset: 0
The top left corner of the array:
1, 2, 6, 24, 120, 720, 5040, 40320
3, 8, 30, 144, 840, 5760, 45360, 403200
11, 38, 174, 984, 6600, 51120, 448560, 4394880
49, 212, 1158, 7584, 57720, 499680, 4843440, 51932160
261, 1370, 8742, 65304, 557400, 5343120, 56775600, 661933440
1631, 10112, 74046, 622704, 5900520, 62118720, 718709040, 9059339520
Topmost row (row 0):
A000142, Row 1:
A001048 (without its initial 2), Row 2:
A001344 (from a(1) = 11 onward), Row 3:
A001345 (from a(1) = 49 onward), Row 4:
A001346 (from a(1) = 261 onward), Row 5:
A001347 (from a(1) = 1631 onward).
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T[r_, c_]:=Sum[Binomial[r, k](1 + c + k)!, {k, 0, r}]; Table[T[c, r - c], {r, 0, 10}, {c, 0, r}] // Flatten (* Indranil Ghosh, Apr 11 2017 *)
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T(r, c) = sum(k=0, r, binomial(r, k)*(1 + c + k)!);
for(r=0, 10, for(c=0, r, print1(T(c, r - c),", ");); print();) \\ Indranil Ghosh, Apr 11 2017
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from sympy import binomial, factorial
def T(r, c): return sum([binomial(r, k) * factorial(1 + c + k) for k in range(r + 1)])
for r in range(11): print([T(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Apr 11 2017
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(define (A276588 n) (A276588bi (A002262 n) (A025581 n)))
(define (A276588bi row col) (A276075 (A066117bi (+ 1 row) (+ 1 col)))) ;; Code for A066117bi given in A066117, and for A276075 under the respective entry.
A076571
Binomial triangle based on factorials.
Original entry on oeis.org
1, 1, 2, 2, 3, 5, 6, 8, 11, 16, 24, 30, 38, 49, 65, 120, 144, 174, 212, 261, 326, 720, 840, 984, 1158, 1370, 1631, 1957, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 13700, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 109601
Offset: 0
Rows start:
1;
1, 2;
2, 3, 5;
6, 8, 11, 16;
24, 30, 38, 49, 65;
120, 144, 174, 212, 261, 326;
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
- E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35. See Table I.
- D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
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A076571:= func< n,k| (&+[Binomial(k,j)*Factorial(n-j): j in [0..k]]) >;
[A076571(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023
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A076571[n_, k_]:= n!*Hypergeometric1F1[-k,-n,1];
Table[A076571[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 05 2023 *)
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def A076571(n,k): return sum(binomial(k,j)*factorial(n-j) for j in range(k+1))
flatten([[A076571(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023
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