A008859 -id:A008859 - cp's OEIS frontend

A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 3, 1, 0, 1, 2, 4, 4, 1, 1, 1, 2, 4, 7, 5, 1, 0, 1, 2, 4, 8, 11, 6, 1, 1, 1, 2, 4, 8, 15, 16, 7, 1, 0, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 0, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 0

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(0,2n)=T(1,n) by T(0,2n)=T(m,n) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058394, A058395, A057884, (and effectively A007318).

			Rows are (1,0,1,0,1,0,1,...), (1,1,1,1,1,1,...), (1,2,2,2,2,2,...), (1,3,4,4,4,...) etc.

Rows are A000035 (A000012 with zeros), A000012, A040000 etc. Columns are A000012, A001477, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863 etc. Diagonals include A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, etc. The triangles A008949, A054143 and A055248 also appear in the half of the array which is not powers of 2.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(1, 1)=1, T(0, 2n)=T(1, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2).

A219615 a(n) = Sum_{k=0..12} binomial(n,k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16369, 32647, 64839, 127858, 249528, 480492, 910596, 1695222, 3096514, 5546382, 9740686, 16777216, 28354132, 47050564, 76717268, 123012781, 194129627, 301766029, 462411533, 699030226, 1043243132

Offset: 0

Mokhtar Mohamed, Nov 23 2012

a(n) is the number of compositions (ordered partitions) of n+1 into thirteen or fewer parts.

a(n) is the sum of the first thirteen terms in the n-th row of Pascal's triangle.

			a(13)= 8191 because there are (2^13) -1 compositions of 14 into thirteen or fewer parts. When 1<= n <= 12, for n=5, a(5) = 2*a(4) = 2*16 = 32. For n=12, a(12) = 2*a(11)= 2*2048 = 4096. When n>12, for n=13, a(13) = 2*a(12) - binomial(12,12) = 2*4096 - 1 = 8191. For n = 15, a(15) = 2*a(14) - binomial(14,12) = 2*16369 - 91 = 32738 - 91 = 32647.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A000127, A006261, A008859, A008860, A008861, A008862, A008863, A219531.

Table[Sum[Binomial[n, k], {k, 0, 12}], {n, 0, 40}] (* T. D. Noe, Nov 27 2012 *)
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{1,2,4,8,16,32,64,128,256,512,1024,2048,4096},40] (* Harvey P. Dale, Nov 29 2012 *)

a(n)=sum(k=1,12,binomial(n,k)) \\ Charles R Greathouse IV, Nov 27 2012

a(n) = (n^12 - 54n^11 + 1397n^10 - 21450n^9 + 218823n^8 - 1508562n^7 + 7374191n^6 - 23551110n^5 + 58206676n^4 - 48306984n^3 + 173699712n^2 + 312888960n)/479001600. - Charles R Greathouse IV, Nov 27 2012

a(0)=1, a(1)=2, a(2)=4, a(3)=8, a(4)=16, a(5)=32, a(6)=64, a(7)=128, a(8)=256, a(9)=512, a(10)=1024, a(11)=2048, a(12)=4096, a(n)= 13*a(n-1)- 78*a(n-2)+286*a(n-3)-715*a(n-4)+1287*a(n-5)-1716*a(n-6)+ 1716*a(n-7)- 1287*a(n-8)+715*a(n-9)-286*a(n-10)+78*a(n-11)-13*a(n-12)+a(n-13). - Harvey P. Dale, Nov 29 2012

Sequence corrected and extended by T. D. Noe, Nov 26 2012

Definition corrected by Harvey P. Dale, Nov 29 2012

A219676 a(n) = Sum_{k=0..13} binomial(n, k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16383, 32752, 65399, 130238, 258096, 507624, 988116, 1898712, 3593934, 6690448, 12236830, 21977516, 38754732, 67108864, 114159428, 190876696, 313889477, 508019104, 809785133, 1272196666

Offset: 0

Mokhtar Mohamed, Nov 24 2012

a(n) is the number of compositions (ordered partitions) of n+1 into fourteen or fewer parts.

a(n) is the sum of the first fourteen terms in the n-th row of Pascal's triangle.

			a(14) = 16383 because there are 2^14 = 16384 compositions of 15 into any size parts but one of the compositions (1 + 1 + ... + 1 = 15) has more than fourteen parts.
When 1 <= n <= 13, a(7) = 2*a(6) = 2*64= 128, a(13) = 2*a(12) = 2*4096 = 8192.
When n > 13, a(14) = 2*a(13) - C(13, 13) = 2*8192 - 1 = 16383, a(15) = 2*a(14) - C(14, 13) = 2*16383 - 14 = 32766 - 14 = 32752.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000127, A006261, A008859, A008860, A008861, A008862, A008863, A219531.

f:= n -> add(binomial(n,k),k=0..13):
map(f, [$0..100]); # Robert Israel, Mar 14 2018

Table[Sum[Binomial[n, k], {k, 0, 13}], {n, 0, 40}] (* T. D. Noe, Nov 26 2012 *)

a(n) = Sum_{k=1..7} binomial(n+1, 2k-1).
a(n) = 1 +(n^13 -65*n^12 +2015*n^11 -37609*n^10 +470613*n^9 -4081935*n^8 +25378925*n^7 -110205667*n^6 +351042406*n^5 -657328100*n^4 +1303568760*n^3 +771653376*n^2 +4546558080*n)/13!. - corrected by Mokhtar Mohamed, Dec 01 2012
G.f.: (1 - 12*x + 67*x^2 - 230*x^3 + 541*x^4 - 920*x^5 + 1163*x^6 - 1106*x^7 + 791*x^8 - 420*x^9 + 161*x^10 - 42*x^11 + 7*x^12)/(1-x)^14.
a(n) = 2*a(n-1), for 1 <= n <= 13, with a(0) = 1, a(n) = 2*a(n-1) - C(n-1, 13), for n > 13.

Corrected and extended by T. D. Noe, Nov 26 2012

A084638 Binomial transform of (1,0,1,0,1,0,1,0,2,0,2,0,2,....).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 129, 265, 558, 1200, 2610, 5682, 12288, 26292, 55587, 116179, 240366, 493108, 1004780, 2036692, 4112144, 8278552, 16631717, 33364381, 66863358, 133903816, 268037862, 536371734, 1073120208, 2146715436, 4294024647, 8588785575

Offset: 0

Paul Barry, Jun 06 2003

The sequence starting 1,2,4,... is the binomial transform of (1,1,1,1,1,1,1,2,2...) with a(n) = Sum_{k=0..6} C(n,k) + 2*Sum_{k=7..n} C(n,k) = 2^(n+1) - A008859(n). This gives the partial sums of A084637.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-35,77,-105,91,-49,15,-2).

Cf. A000225, A000325, A008859, A084634, A084635, A084636, A084637.

[2^n -4 -(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160)/720 + 0^n: n in [0..50]]; // G. C. Greubel, Mar 20 2023

Table[2^n -4 -(1/6!)*(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160) + Boole[n==0], {n,0,50}] (* G. C. Greubel, Mar 20 2023 *)

Vec((1-8*x+28*x^2-56*x^3+70*x^4-56*x^5+28*x^6-8*x^7+2*x^8)/((1-x)^7*(1-2*x)) + O(x^50)) \\ Colin Barker, Mar 17 2016

[2^n -4 -(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160)/720 + 0^n for n in range(51)] # G. C. Greubel, Mar 20 2023

a(n) = Sum_{k=0..3, C(n, 2*k)} + 2*Sum_{k=4..floor(n/2), C(n, 2*k)}.
a(n) = (n^6-15*n^5+115*n^4-405*n^3+964*n^2-660*n+720)/720 + 2*Sum_{k=4..floor(n/2), C(n, 2k)}.
G.f.: (1-8*x+28*x^2-56*x^3+70*x^4-56*x^5+28*x^6-8*x^7+2*x^8) / ((1-x)^7*(1-2*x)). - Colin Barker, Mar 17 2016

A220051 Sum_{k=0..14} binomial(n,k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32767, 65519, 130918, 261156, 519252, 1026876, 2014992, 3913704, 7507638, 14198086, 26434916, 48412432, 87167164, 154276028, 268435456, 459312152, 773201629, 1281220733, 2091005866

Offset: 0

Mokhtar Mohamed, Dec 03 2012

a(n) is the number of compositions (ordered partitions) of n+1 into fifteen or fewer parts.
a(n) = sum(binomial(n+1,2k), for k = 0..7).
a(n) is the sum of the first fifteen terms in the n-th row of Pascal's triangle.

			a(15) = 32767 because there are 2^15 = 32768 compositions of 16 into any size parts but one of the compositions (1 + 1 + ... + 1 = 16) has more than fifteen parts.
When 1 <= n <= 14, for n=10, a(10) = 2*a(9) = 2*512 = 1024. For n=14, a(14) = 2*a(13) = 2*8192 = 16384.
When n > 14, for n = 15, a(15) = 2*a(14) -C(14,14) = 2*16384 -1 = 32767. For n=20, a(20) = 2*a(19) -C(19,14) = 2*519252 -11626 = 1038504 -11626 = 1026876.

Indranil Ghosh, Table of n, a(n) for n = 0..1000

Cf. A000127, A006261, A008859, A008860, A008861, A008862, A008863, A219531, A219615, A219676.

Table[Sum[Binomial[n,k],{k,0,14}],{n,0,33}] (* Indranil Ghosh, Feb 22 2017 *)
NestList[{#1 + 1, 2 #2 - Boole[#1 >= 14] Binomial[#1, 14]} & @@ # &, {0, 1}, 33][[All, -1]] (* Michael De Vlieger, Feb 22 2017 *)

a(n)=sum(k=0,14,binomial(n,k)) \\ Indranil Ghosh, Feb 23 2017

a(n) = 1 + (n^14 - 77*n^13 + 2821*n^12 - 6288*n^11 + 947947*n^10 - 10081071*n^9 + 77889383*n^8 - 435638203*n^7 + 1793239448*n^6 - 5043110072*n^5 + 1111159696*n^4 - 8346754416*n^3 + 30605906304*n^2 + 57424792320*n)/14!.
G.f.: (1 - 13x + 79x^2 - 297x^3 + 771x^4 - 1461x^5 + 2083x^6 - 2269x^7 + 1897x^8 - 1211x^9 + 581x^10 - 203x^11 + 49x^12 - 7x^13 + x^14)/(1-x)^15.
a(n) = 2*a(n-1), for 1 <= n <= 14, with a(0) = 1, a(n) = 2*a(n-1) - C(n-1,14), for n> 14.

A362194 Number of Grassmannian permutations of size n that avoid a pattern, sigma, where sigma is a pattern of size 7 with exactly one descent.

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 58, 120, 239, 457, 838, 1475, 2498, 4083, 6462, 9934, 14877, 21761, 31162, 43777, 60440, 82139, 110034, 145476, 190027, 245481, 313886, 397567, 499150, 621587, 768182, 942618, 1148985, 1391809, 1676082, 2007293, 2391460, 2835163, 3345578, 3930512

Offset: 0

Jessica A. Tomasko, Apr 20 2023

A permutation is said to be Grassmannian if it has at most one descent. The definition for sigma is a pattern of size 7 with exactly one descent. For example, sigma can be chosen to be 1247356, 2413567, 3671245, 5712346, etc.

Juan B. Gil and Jessica Tomasko, Restricted Grassmannian permutations, ECA 2:4 (2022) Article S4PP6.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000325, A008859, A362193.

a(n) = 1 + sum(i=2, 6, binomial(n,i)) \\ Andrew Howroyd, Apr 20 2023

a(n) = 1 + Sum_{i=2..6} binomial(n, i).
a(n) = A008859(n) - n.
G.f.: (1-6*x+16*x^2-23*x^3+19*x^4-8*x^5+2*x^6)/(1-x)^7.
E.g.f.: exp(x)*(720 + 360*x^2 + 120*x^3 + 30*x^4 + 6*x^5 + x^6)/720. - Stefano Spezia, Apr 20 2023

A373005 Array read by ascending antidiagonals: A(n,k) is the maximum possible cardinality of a set of points of diameter at most k-1 in {0,1}^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 2, 2, 0, 0, 1, 2, 3, 2, 1, 0, 1, 2, 4, 4, 2, 2, 0, 1, 2, 5, 6, 4, 2, 1, 0, 1, 2, 6, 8, 7, 4, 2, 0, 0, 1, 2, 7, 10, 11, 8, 4, 2, 1, 0, 1, 2, 8, 12, 16, 14, 8, 4, 2, 2, 0, 1, 2, 9, 14, 22, 22, 15, 8, 4, 2, 1, 0, 1, 2, 10, 16, 29, 32, 26, 16, 8, 4, 2, 0

Offset: 0

Stefano Spezia, May 19 2024

A(n,k) is also the size of the Hamming ball in {0,1}^n of radius (k-1)/2 if k is odd and of the union of two Hamming balls in {0,1}^n of radius k/2-1 whose centers are of Hamming distance 1 if k is even.

			The array begins:
  1, 1, 2, 1,  0,  1,  2,  1, ...
  0, 1, 2, 2,  2,  2,  2,  2, ...
  0, 1, 2, 3,  4,  4,  4,  4, ...
  0, 1, 2, 4,  6,  7,  8,  8, ...
  0, 1, 2, 5,  8, 11, 14, 15, ...
  0, 1, 2, 6, 10, 16, 22, 26, ...
  0, 1, 2, 7, 12, 22, 32, 42, ...
  0, 1, 2, 8, 14, 29, 44, 64, ...
  ...

Noga Alon, Zhihan Jin, and Benny Sudakov, The Helly number of Hamming balls and related problems, arXiv:2405.10275 [math.CO], 2024. See p. 3.
S. L. Bezrukov, Specification of all maximal subsets of the unit cube with respect to given diameter. Problemy Peredachi Informatsii, pages 106-109, 1987. On ResearchGate.
G. Katona, Intersection theorems for systems of finite sets, Acta Mathematica Academiae Scientiarum Hungaricae 15, 329-337 (1964).
Daniel J. Kleitman, On a combinatorial conjecture of Erdös, Journal of Combinatorial Theory, Series A 1, 209-214, (1966).

Cf. A000007 (k=0), A000012 (k=1), A000124 (k=5), A000125 (k=7), A005843 (k=4), A006261 (k=11), A007395 (k=2), A008859 (k=13), A011782 (main diagonal), A014206, A046127 (k=8), A059173, A059174, A130130 (n=1), A158411 (n=2), A373006 (antidiagonal sums).

A[n_,k_]:=If[OddQ[k],Sum[Binomial[n,i],{i,0,(k-1)/2}], Binomial[n-1,k/2-1]+Sum[Binomial[n,i],{i,0,k/2-1}]]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten

A(n,k) = Sum_{i=0..(k-1)/2} binomial(n,i) if k is odd;
A(n,k) = binomial(n-1,k/2-1) + Sum_{i=0..k/2-1} binomial(n,i) if k is even.
A(n,3) = n+1.
A(n,6) = A014206(n-1).
A(n,9) = A000127(n+1).
A(n,10) = A059173(n) for n > 0.
A(n,12) = A059174(n) for n > 0.
A(0,k) = A007877(k) for k > 0.

cp's OEIS Frontend

A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.

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A219615 a(n) = Sum_{k=0..12} binomial(n,k).

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A219676 a(n) = Sum_{k=0..13} binomial(n, k).

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A084638 Binomial transform of (1,0,1,0,1,0,1,0,2,0,2,0,2,....).

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A220051 Sum_{k=0..14} binomial(n,k).

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A362194 Number of Grassmannian permutations of size n that avoid a pattern, sigma, where sigma is a pattern of size 7 with exactly one descent.

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A373005 Array read by ascending antidiagonals: A(n,k) is the maximum possible cardinality of a set of points of diameter at most k-1 in {0,1}^n.

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