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A055248
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Triangle of partial row sums of triangle A007318(n,m) (Pascal's triangle). Triangle A008949 read backwards.
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23
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1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 11, 5, 1, 32, 31, 26, 16, 6, 1, 64, 63, 57, 42, 22, 7, 1, 128, 127, 120, 99, 64, 29, 8, 1, 256, 255, 247, 219, 163, 93, 37, 9, 1, 512, 511, 502, 466, 382, 256, 130, 46, 10, 1, 1024, 1023, 1013, 968, 848, 638, 386, 176, 56, 11, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is 1/((1-2*z)*(1-x*z/(1-z))).
a(n,m) = A008949(n,n-m), if n>m >= 0.
Binomial transform of the all 1's triangle: as a Riordan array, it factors to give (1/(1-x),x/(1-x))(1/(1-x),x). Viewed as a number square read by anti-diagonals, it has T(n,k)=sum{j=0..n, C(n+k,n-j)} and is then the binomial transform of the Whitney square A004070. - Paul Barry (pbarry(AT)wit.ie), Feb 03 2005
Riordan array (1/(1-2x), x/(1-x)). Diagonal sums are A027934. - Paul Barry (pbarry(AT)wit.ie), Jan 30 2005
Eigensequence of the triangle = A005493: (1, 3, 10, 37, 151, 674,...); row sums of triangles A011971 and A159573. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 16 2009]
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FORMULA
| a(n, m)=sum(A007318(n, k), k=m..n), (partial row sums in columns m).
Column m recursion: a(n, m)= sum(a(j, m), j=m..n-1)+ A007318(n, m) if n >= m >= 0, a(n, m) := 0 if n<m.
G.f. for column m: (1/(1-2*x))*(x/(1-x))^m, m >= 0.
T(n, k)=sum{j=0..n, binomial(n, k+j)}. - Paul Barry (pbarry(AT)wit.ie), Jan 30 2005
a(n, m)=sum{j=0..n, binomial(n, m+j)} - Paul Barry (pbarry(AT)wit.ie), Feb 03 2005
Inverse binomial transform (by columns) of A112626. - Ross La Haye (rlahaye(AT)new.rr.com), Dec 31 2006
T(2n,n)=A032443(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 16 2009]
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EXAMPLE
| {1}; {2,1}; {4,3,1}; {8,7,4,1};...
Fourth row polynomial (n=3): p(3,x)= 8+7*x+4*x^2+x^3
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CROSSREFS
| Cf. A008949, A007318. Column sequences: A000079 (powers of 2), A000225, A000295, A002662-4, A035038-42 for m=0..10, Row sums: A001792(n) = A055249(n, 0).
A011971, A159573 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 16 2009]
Sequence in context: A109435 A134392 A048483 * A103316 A140069 A105851
Adjacent sequences: A055245 A055246 A055247 * A055249 A055250 A055251
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 26 2000
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