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A074909
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Running sum of Pascal's triangle (A007318), or beheaded Pascal triangle read by beheaded rows.
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14
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1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 6, 15, 20, 15, 6, 1, 7, 21, 35, 35, 21, 7, 1, 8, 28, 56, 70, 56, 28, 8, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 12, 66, 220, 495, 792, 924
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| This sequence counts the "almost triangular" partitions of n. A partition is triangular if it is of the form 0+1+2+...+k. Examples 3=0+1+2, 6=0+1+2+3. An "almost triangular" partition is a triangular partition with at most 1 added to each of the parts. Examples: 7=1+1+2+3=0+2+2+3=0+1+3+3=0+1+2+4. Thus a(7)=4. 8=1+2+2+3=1+1+3+3=1+1+2+4=0+2+3+3=0+2+2+4=0+1+3+4 so a(8)=6. - Moshe Newman (moshneumann(AT)hotmail.com), Dec 19 2002
The "almost triangular" partitions are the ones cycled by the operation of "bulgarian solitaire", as defined by Martin Gardner.
Also the number of increasing acyclic functions from {1,...,n-k+1} to {1,2,...,n+2}. A function f is acyclic if for every subset B of the domain the image of B under f does not equal B. For example, T(3,1)=4 since there are exactly 4 increasing acyclic functions from {1,2,3} to {1,2,3,4,5}: f1={(1,2),(2,3),(3,4)}, f2={(1,2),(2,3),(3,5)}, f3={(1,2),(2,4),(3,5)} and f4={(1,3),(2,4),(4,5)}. - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Mar 14 2008
Second Bernoulli polynomials are (from A164555 instead of A027641) B2(n,x)= 1, 1/2,1, 1/6,1,1, 0,1/2,3/2,1, -1/30,0,1,2,1, 0,-1/6,0,5/3,5/2,1,... Then (B2(n,x)/A002260)= 1, 1/2,1/2, 1/6,1/2,1/3, 0,1/4,1/2,1/4, -1/30,0,1/3,1/2,1/5, 0,-1/12,0,5/12,1/2,1/6, .See (from Faulhaber 1631) Jacob Bernoulli Summae Potestatum (sum of powers) in A159688. Inverse polynomials are 1, -1,2, 1,-3,3, -1,4,-6,4, = A074909 with negative even diagonals. Reflected A053382/A053383 = reflected B(n,x)=RB(n,x)=1, -1/2,1, 1/6,-1,1, 0,1/2,-3/2,1, . A074909 is inverse of RB(n,x)/A002260 = 1, -1/2,1/2, 1/6,-1/2,1/3, 0,1/4,-1/2,1/4, . [From Paul Curtz (bpcrtz(AT)free.fr), Jun 21 2010]
A054143 is the fission of the polynomial sequence (p(n,x)) given by p(n,x)=x^n+x^(n-1)+...+x+1 by the polynomial sequence ((x+1)^n). See A193842 for the definition of fission. [From Clark Kimberling, Aug 7 2011]
Reversal of A135278. - DELEHAM Philippe, Feb 11 2012
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LINKS
| J. R. Griggs, , bulgarian solitaire partitions
D. P. Walsh, A short note on increasing acyclic functions.
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FORMULA
| T(n, k)=Sum_{i=0..n} C(i, n-k) = C(n+1, k).
Row k has g.f. (1+x)^(k+1)-x^(k+1).
T(n, k)=T(n-1, k-1)+T(n-1, k) for k: 0<k<n, T(n, 0)=1, T(n, n)=n. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 18 2005
Start with A007318 - I, then delete right border of zeros; (I = Identity matrix). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2007
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EXAMPLE
| T(4,2)=10 = 0+0+1+3+6.
{1}, {1,2}, {1,3,3}, {1,4,6,4}, {1,5,10,10,5},...
Triangle begins:
1
1, 2
1, 3, 3
1, 4, 6, 4
1, 5, 10, 10, 5
1, 6, 15, 20, 15, 6
1, 7, 21, 35, 35, 21, 7
1, 8, 28, 56, 70, 56, 28, 8
1, 9, 36, 84, 126, 126, 84, 36, 9
1, 10, 45, 120, 210, 252, 210, 120, 45, 10
1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11
1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12
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MAPLE
| for n from 0 to 11 do seq(binomial(n+1, k), k =0..n); od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 09 2006
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MATHEMATICA
| Table[Sum[Binomial[k, m], {k, 0, n}], {n, 0, 12}, {m, 0, n}] or Table[ Binomial[n, m], {n, 1, 12}, {m, 1, n}]
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CROSSREFS
| Cf. A007318. Row sums are A000225, diagonal sums are A052952.
Cf. A007318.
The number of acyclic functions is A058127.
Sequence in context: A131251 A057145 A134394 * A135278 A034356 A075195
Adjacent sequences: A074906 A074907 A074908 * A074910 A074911 A074912
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KEYWORD
| easy,nonn,tabl,changed
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AUTHOR
| Wouter Meeussen (wouter.meeussen(AT)pandora.be), Oct 01 2002
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EXTENSIONS
| I added an initial 1 at the suggestion of Paul Barry, which makes the triangle a little nicer but may mean that some of the formulae will now need adjusting. - N. J. A. Sloane (njas(AT)research.att.com), Feb 11 2003
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