A002054 -id:A002054 - cp's OEIS frontend

A108044 Triangle read by rows: right half of Pascal's triangle (A007318) interspersed with 0's.

1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 6, 0, 4, 0, 1, 0, 10, 0, 5, 0, 1, 20, 0, 15, 0, 6, 0, 1, 0, 35, 0, 21, 0, 7, 0, 1, 70, 0, 56, 0, 28, 0, 8, 0, 1, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1, 252, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1, 0, 462, 0, 330, 0, 165, 0, 55, 0, 11, 0, 1, 924, 0, 792, 0, 495, 0, 220, 0, 66

Offset: 0

N. J. A. Sloane, Jun 02 2005

Column k has e.g.f. Bessel_I(k,2x). - Paul Barry, Mar 10 2010
T(n,k) is the number of binary sequences of length n in which the number of ones minus the number of zeros is k. If 2 divides(n+k), such a sequence will have (n+k)/2 ones and (n-k)/2 zeros. Since there are C(n,(n+k)/2) ways to choose the sequence entries that get a one, T(n,k)=binomial(n,(n+k)/2) whenever (n+k) is even and T(n,k)= 0 otherwise. See the example below in the example section. - Dennis P. Walsh, Apr 11 2012
T(n,k) is the number of walks on the semi-infinite integer line with n steps that end at k. The walks start at 0, move at each step either one to the left or one to the right, and never enter the region of negative k. [Walks with impenetrable wall at -1/2. Dyck excursions of n steps that end at level k.] The variant without the restriction of negative positions is A053121. - R. J. Mathar, Nov 02 2023

			Triangle begins:
  1
  0 1
  2 0 1
  0 3 0 1
  6 0 4 0 1
  0 10 0 5 0 1
  20 0 15 0 6 0 1
From _Paul Barry_, Mar 10 2010: (Start)
Production matrix is
  0, 1,
  2, 0, 1,
  0, 1, 0, 1,
  0, 0, 1, 0, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End)
T(5,1)=10 since there are 10 binary sequences of length 5 in which the number of ones exceed the number of zeros by exactly 1, namely, 00111, 01011, 01101, 01110, 10011, 10101, 10110, 11001, 11010, and 11100. Also, T(5,2)=0 since there are no binary sequences in which the number of ones exceed the number of zeros by exactly 2. - _Dennis P. Walsh_, Apr 11 2012

Harvey P. Dale, Rows n = 0..125 of triangle, flattened
Paul Barry and Aoife Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 8.
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 14.
Robert S. Maier, Sheffer Polynomials and the s-ordering of Exponential Boson Operators, arXiv:2508.13094 [quant-ph], 2025. See p. 27.
Paul Peart and Wen-Jin Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
Louis W. Shapiro, Seyoum Getu, Wen-Jin Woan, and Leon C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A108045 (matrix inverse),

Cf. A204293, A357136, A000984 (column 0), A001700 (column 1), A001791 (column 2), A002054 (column 3)

import Data.List (intersperse)
a108044 n k = a108044_tabl !! n !! k
a108044_row n = a108044_tabl !! n
a108044_tabl = zipWith drop [0..] $ map (intersperse 0) a007318_tabl
-- Reinhard Zumkeller, May 18 2013

T:=proc(n,k) if n+k mod 2 = 0 then binomial(n,(n+k)/2) else 0 fi end: for n from 0 to 13 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Jun 05 2005

b[n_,k_]:=If[EvenQ[n+k],Binomial[n,(n+k)/2],0]; Flatten[Table[b[n,k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, May 05 2013 *)

Each entry is the sum of those in the previous row that are to its left and to its right.
Riordan array (1/sqrt(1-4*x^2), (1-sqrt(1-4*x^2))/(2*x)).
T(n, k) = binomial(n, (n+k)/2) if n+k is even, T(n, k)=0 if n+k is odd. G.f.=f/(1-tg), where f=1/sqrt(1-4x^2) and g=(1-sqrt(1-4x^2))/(2x). - Emeric Deutsch, Jun 05 2005
From Peter Bala, Jun 29 2015: (Start)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - sqrt(1 - 4*x) )/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + x^2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ).
The inverse array is A108045 (a hitting time array with h(x) = x/(1 + x^2)). (End)

More terms from Emeric Deutsch and Christian G. Bower, Jun 05 2005

A146305 Array T(n,m) = 2(2m+3)!(4n+2m+1)!/(m!(m+2)!n!*(3n+2m+3)!) read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 13, 20, 21, 14, 68, 100, 105, 84, 42, 399, 570, 595, 504, 330, 132, 2530, 3542, 3675, 3192, 2310, 1287, 429, 16965, 23400, 24150, 21252, 16170, 10296, 5005, 1430, 118668, 161820, 166257, 147420, 115500, 78936, 45045, 19448, 4862, 857956

Offset: 0

R. J. Mathar, Oct 29 2008

T(n,m) is the number of rooted nonseparable (2-connected) triangulations of the disk with n internal nodes and 3 + m nodes on the external face. The triangulation has 2*n + m + 1 triangles and 3*(n+1) + 2*m edges. - Andrew Howroyd, Feb 21 2021

			The array starts at row n=0 and column m=0 as
.....1......2.......5......14.......42.......132
.....1......5......21......84......330......1287
.....3.....20.....105.....504.....2310.....10296
....13....100.....595....3192....16170.....78936
....68....570....3675...21252...115500....602316
...399...3542...24150..147420...844074...4628052
..2530..23400..166257.1057224..6301680..35939904
.16965.161820.1186680.7791168.47948670.282285432

Andrew Howroyd, Table of n, a(n) for n = 0..1325
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.

Columns m=0..3 are A000260, A197271(n+1), A341853, A341854.
Rows n=0..2 are A000108(n+1), A002054(n+1) and A000917.
Antidiagonal sums are A000260(n+1).
Cf. A169808 (unrooted), A169809 (achiral), A262586 (oriented).

A146305 := proc(n,m)
    2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)! ;
end proc:
for d from 0 to 13 do
    for m from 0 to d do
        printf("%d,", A146305(d-m,m)) ;
    end do:
end do:

T[n_, m_] := 2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)!; Table[T[n-m, m], {n, 0, 13}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 06 2014, after Maple *)

T(n,m)={2*(2*m+3)!*(4*n+2*m+1)!/(m!*(m+2)!*n!*(3*n+2*m+3)!)} \\ Andrew Howroyd, Feb 21 2021

A371965 a(n) is the sum of all peaks in the set of Catalan words of length n.

Original entry on oeis.org

0, 0, 0, 1, 6, 27, 111, 441, 1728, 6733, 26181, 101763, 395693, 1539759, 5997159, 23381019, 91244934, 356427459, 1393585779, 5453514729, 21358883439, 83718027429, 328380697629, 1288947615849, 5062603365999, 19896501060225, 78239857877649, 307831771279549, 1211767933187601

Offset: 0

Stefano Spezia, Apr 14 2024

nonn

			a(3) = 1 because there is 1 Catalan word of length 3 with one peak: 010.
a(4) = 6 because there are 6 Catalan words of length 4 with one peak: 0010, 0100, 0101, 0110, 0120, and 0121 (see Figure 10 at p. 19 in Baril et al.).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See Corollary 4.7, p. 19.

Cf. A371963, A371964.

Cf. A002054, A275289.

a:= proc(n) option remember; `if`(n<3, 0,
      a(n-1)+binomial(2*n-3, n-3))
    end:
seq(a(n), n=0..28);  # Alois P. Heinz, Apr 15 2024
# Second Maple program:
A371965 := series((exp(2*x)*BesselI(0,2*x)-1)/2-exp(x)*(int(BesselI(0,2*x)*exp(x), x)), x = 0, 29):
seq(n!*coeff(A371965, x, n), n = 0 .. 28); # Mélika Tebni, Jun 15 2024

CoefficientList[Series[(1-3x-(1-x)Sqrt[1-4x])/(2(1-x) Sqrt[1-4x]),{x,0,28}],x]

from math import comb
def A371965(n): return sum(comb((n-i<<1)-3,n-i-3) for i in range(n-2)) # Chai Wah Wu, Apr 15 2024

G.f.: (1 - 3*x - (1 - x)*sqrt(1 - 4*x))/(2*(1 - x)*sqrt(1 - 4*x)).
a(n) = Sum_{i=1..n-1} binomial(2*(n-i)-1,n-i-2).
a(n) ~ 2^(2*n)/(6*sqrt(Pi*n)).
a(n)/A371963(n) ~ 1.
a(n) - a(n-1) = A002054(n-2).
From Mélika Tebni, Jun 15 2024: (Start)
E.g.f.: (exp(2*x)*BesselI(0,2*x)-1)/2 - exp(x)*Integral_{x=-oo..oo} BesselI(0,2*x)*exp(x) dx.
a(n) = binomial(2*n,n)*(1/2 + hypergeom([1,n+1/2],[n+1],4)) + i/sqrt(3) - 0^n/2.
a(n) = (3*A106191(n) + A006134(n) + 4*0^n) / 8.
a(n) = A281593(n) - (A000984(n) + 0^n) / 2. (End)
Binomial transform of A275289. - Alois P. Heinz, Jun 20 2025

A133336 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 5, 5, 1, 0, 14, 21, 9, 1, 0, 42, 84, 56, 14, 1, 0, 132, 330, 300, 120, 20, 1, 0, 429, 1287, 1485, 825, 225, 27, 1, 0, 1430, 5005, 7007, 5005, 1925, 385, 35, 1, 0, 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1, 0, 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1, 0

Offset: 0

Philippe Deléham, Oct 19 2007

Mirror image of triangle A086810; another version of A126216.
Equals A131198*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 23 2007
Diagonal sums: A119370. - Philippe Deléham, Nov 09 2009

			Triangle begins:
    1;
    1,    0;
    2,    1,    0;
    5,    5,    1,   0;
   14,   21,    9,   1,   0;
   42,   84,   56,  14,   1,  0;
  132,  330,  300, 120,  20,  1, 0;
  429, 1287, 1485, 825, 225, 27, 1, 0;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.

Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281, A033282.

[[Binomial(n-1,k)*Binomial(2*n-k,n)/(n+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 05 2018

Table[Binomial[n-1,k]*Binomial[2*n-k,n]/(n+1), {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 05 2018 *)

for(n=0,10, for(k=0,n, print1(binomial(n-1,k)*binomial(2*n-k,n)/(n+1), ", "))) \\ G. C. Greubel, Feb 05 2018

Sum_{k=0..n} T(n,k)*x^k = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Nov 05 2007
Sum_{k=0..n} T(n,k)*(-2)^k*5^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
T(n,k) = binomial(n-1,k)*binomial(2n-k,n)/(n+1), k <= n. - Philippe Deléham, Nov 02 2009

A345960 Numbers whose prime indices have alternating sum 2.

Original entry on oeis.org

3, 12, 27, 30, 48, 70, 75, 108, 120, 147, 154, 192, 243, 270, 280, 286, 300, 363, 432, 442, 480, 507, 588, 616, 630, 646, 675, 750, 768, 867, 874, 972, 1080, 1083, 1120, 1144, 1200, 1323, 1334, 1386, 1452, 1470, 1587, 1728, 1750, 1768, 1798, 1875, 1920, 2028

Offset: 1

Gus Wiseman, Jul 12 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.
Also numbers with odd Omega (A001222) and exactly two odd conjugate prime indices. The version for even Omega is A345962, and the union is A345961. Conjugate prime indices are listed by A321650 and ranked by A122111.

			The initial terms and their prime indices:
    3: {2}
   12: {1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   48: {1,1,1,1,2}
   70: {1,3,4}
   75: {2,3,3}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
  147: {2,4,4}
  154: {1,4,5}
  192: {1,1,1,1,1,1,2}
  243: {2,2,2,2,2}
  270: {1,2,2,2,3}
  280: {1,1,1,3,4}
  286: {1,5,6}
  300: {1,1,2,3,3}

These partitions are counted by A000097.
The k = 0 version is A000290, counted by A000041.
The k = 1 version is A001105 (reverse: A345958).
The k > 0 version is A026424.
These multisets are counted by A120452.
These are the positions of 2's in A316524 (reverse: A344616).
The k = -1 version is A345959.
The version for reversed alternating sum is A345961.
The k = -2 version is A345962.
A000984/A345909/A345911 count/rank compositions with alternating sum 1.
A002054/A345924/A345923 count/rank compositions with alternating sum -2.
A056239 adds up prime indices, row sums of A112798.
A088218/A345925/A345922 count/rank compositions with alternating sum 2.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 and A325535 count separable and inseparable partitions.
A344606 counts alternating permutations of prime indices.
Cf. A000037, A000070, A028260, A035363, A239830, A341446, A344609, A344610, A344651, A344741, A345197.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[primeMS[#]]==2&]

A030054 a(n) = binomial(2n+1,n-4).

Original entry on oeis.org

1, 11, 78, 455, 2380, 11628, 54264, 245157, 1081575, 4686825, 20030010, 84672315, 354817320, 1476337800, 6107086800, 25140840660, 103077446706, 421171648758, 1715884494940, 6973199770790, 28277527346376, 114456658306760, 462525733568080, 1866442158555975

Offset: 4

N. J. A. Sloane

nonn

Robert Israel, Table of n, a(n) for n = 4..1661
Milan Janjic, Two Enumerative Functions.

Diagonal 10 of triangle A100257.
Fifth unsigned column (s=4) of A113187. - Wolfdieter Lang, Oct 19 2012
Cf. A001622.
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A003516 (m = 5), A002696 (m = 6), A030053 - A030056, A004310 - A004318.

seq(binomial(2*n+1,n-4),n=4..50); # Robert Israel, Jun 11 2019

Table[Binomial[2n+1,n-4],{n,4,40}]  (* Harvey P. Dale, Mar 31 2011 *)

vector(30, n, m=n+4; binomial(2*m+1,m-4)) \\ Michel Marcus, Aug 11 2015

G.f.: x^4*512/((1-sqrt(1-4*x))^9*sqrt(1-4*x))+(-1/x^5+7/x^4-15/x^3+10/x^2-1/x). - Vladimir Kruchinin, Aug 11 2015
From Robert Israel, Jun 11 2019: (Start)
(54 + 36*n)*a(n) + (-438 - 129*n)*a(n + 1) + (714 + 138*n)*a(n + 2) + (-432 - 63*n)*a(n + 3) + (110 + 13*n)*a(n + 4) + (-10 - n)*a(n + 5) = 0.
a(n) ~ 2^(2*n+1)/sqrt(n*Pi). (End)
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=4} 1/a(n) = 317/210 - 2*Pi/(9*sqrt(3)).
Sum_{n>=4} (-1)^n/a(n) = 2908*log(phi)/(5*sqrt(5)) - 8697/70, where phi is the golden ratio (A001622). (End)
G.f.: 2F1([11/2,5],[10],4*x). - Karol A. Penson, Apr 24 2024
From Peter Bala, Oct 13 2024: (Start)
a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * sqrt(x)*(x^4 - 9*x^3 + 27*x^2 - 30*x + 9)/sqrt((4 - x)).
G.f. x^4 * B(x) * C(x)^9, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
D-finite with recurrence -(n+5)*(n-4)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A082971 Number of permutations of {1,2,...,n} containing exactly 3 occurrences of the 132 pattern.

Original entry on oeis.org

1, 14, 82, 410, 1918, 8657, 38225, 166322, 716170, 3059864, 12994936, 54924212, 231235054, 970347575, 4060697955, 16952812170, 70629116910, 293720506860, 1219498444500, 5055891511980, 20933654593020, 86571545598642, 357628915621698, 1475896409177780

Offset: 4

Benoit Cloitre, May 27 2003

			a(4)=1 because we have 1432 (the 132 occurrences are 143, 142 and 132).

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.
Miklós Bóna, Permutations with one or two 132-subsequences, Discrete Math., 181 (1998) 267-274.
T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.

Cf. A002054, A082970, A138162, A138163.

Column k=3 of A263771.

[1] cat [(n^6+51*n^5-407*n^4-99*n^3+7750*n^2-22416*n+20160)* Factorial(2*n-9)/(6*Factorial(n)*Factorial(n-5)): n in [5..30]]; // Vincenzo Librandi, Oct 30 2018

P:=2*x^3-5*x^2+7*x-2: Q:=-22*x^6-106*x^5+292*x^4-302*x^3+135*x^2-27*x+2: g:= (P+Q/(1-4*x)^(5/2))*1/2: gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=4..25); # Emeric Deutsch, Mar 27 2008

a[4] = 1; a[n_] := (n^6 + 51 n^5 - 407 n^4 - 99 n^3 + 7750 n^2 - 22416 n + 20160) (2 n - 9)!/(6 n! (n - 5)!);
Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Oct 30 2018 *)

a(n)=(2*n-9)!/n!/6/(n-5)!*(n^6+51*n^5-407*n^4-99*n^3 +7750*n^2 -22416*n+20160)

a(n) = (2*n-9)!/n!/6/(n-5)! *(n^6+51*n^5-407*n^4-99*n^3 +7750*n^2 -22416*n +20160).

a(n) = (n^6 + 51*n^5 - 407*n^4 - 99*n^3 + 7750*n^2 - 22416*n + 20160)*(2*n-9)!/(6*n!*(n-5)!) for n>=5; a(4)=1. G.f.: (1/2)*(P(x) + Q(x)/(1-4*x)^(5/2)), where P(x) = 2*x^3 - 5*x^2 + 7*x - 2, Q(x) = -22*x^6 - 106*x^5 + 292*x^4 - 302*x^3 + 135*x^2 - 27*x + 2. - Emeric Deutsch, Mar 27 2008

Edited by N. J. A. Sloane, May 21 2008 at the suggestion of R. J. Mathar

A345907 Triangle giving the main antidiagonals of the matrices counting integer compositions by length and alternating sum (A345197).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 0, 4, 3, 1, 1, 0, 0, 3, 6, 4, 1, 1, 0, 0, 6, 9, 8, 5, 1, 1, 0, 0, 0, 18, 18, 10, 6, 1, 1, 0, 0, 0, 10, 36, 30, 12, 7, 1, 1, 0, 0, 0, 20, 40, 60, 45, 14, 8, 1, 1, 0, 0, 0, 0, 80, 100, 90, 63, 16, 9, 1, 1

Offset: 0

Gus Wiseman, Jul 26 2021

The matrices (A345197) count the integer compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2. Here, the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Problem: What are the column sums? They appear to match A239201, but it is not clear why.

			Triangle begins:
   1
   1   1
   0   1   1
   0   1   1   1
   0   2   2   1   1
   0   0   4   3   1   1
   0   0   3   6   4   1   1
   0   0   6   9   8   5   1   1
   0   0   0  18  18  10   6   1   1
   0   0   0  10  36  30  12   7   1   1
   0   0   0  20  40  60  45  14   8   1   1
   0   0   0   0  80 100  90  63  16   9   1   1
   0   0   0   0  35 200 200 126  84  18  10   1   1
   0   0   0   0  70 175 400 350 168 108  20  11   1   1
   0   0   0   0   0 350 525 700 560 216 135  22  12   1   1

Row sums are A163493.
Rows are the antidiagonals of the matrices given by A345197.
The main diagonals of A345197 are A346632, with sums A345908.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
Other diagonals are A008277 of A318393 and A055884 of A320808.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A007318, A008549, A025047, A114121, A344610.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{n-k}],k==(n+ats[#])/2-1&]],{k,0,n-1}],{n,0,15}]

A345962 Numbers whose prime indices have alternating sum -2.

Original entry on oeis.org

10, 21, 40, 55, 84, 90, 91, 160, 187, 189, 210, 220, 247, 250, 336, 360, 364, 391, 462, 490, 495, 525, 551, 640, 713, 748, 756, 810, 819, 840, 858, 880, 988, 1000, 1029, 1073, 1155, 1210, 1271, 1326, 1344, 1375, 1440, 1456, 1564, 1591, 1683, 1690, 1701, 1848

Offset: 1

Gus Wiseman, Jul 12 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.
Also numbers with even Omega (A001222) and exactly two odd conjugate prime indices. The case of odd Omega is A345960, and the union is A345961.

			The initial terms and their prime indices:
   10: {1,3}
   21: {2,4}
   40: {1,1,1,3}
   55: {3,5}
   84: {1,1,2,4}
   90: {1,2,2,3}
   91: {4,6}
  160: {1,1,1,1,1,3}
  187: {5,7}
  189: {2,2,2,4}
  210: {1,2,3,4}
  220: {1,1,3,5}
  247: {6,8}
  250: {1,3,3,3}
  336: {1,1,1,1,2,4}
  360: {1,1,1,2,2,3}

Below we use k to indicate alternating sum.
The k = 0 version is A000290, counted by A000041.
The k = 1 version is A001105 (reverse: A345958).
The k > 0 version is A026424.
These are the positions of -2's in A316524.
These multisets are counted by A344741 (positive 2: A120452).
The k = -1 version is A345959.
The k = 2 version is A345960, counted by A000097.
A002054/A345924/A345923 count/rank compositions with alternating sum -2.
A056239 adds up prime indices, row sums of A112798.
A088218/A345925/A345922 count/rank compositions with alternating sum 2.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 and A325535 count separable and inseparable partitions.
A344606 counts alternating permutations of prime indices.
Cf. A000037, A000070, A001791, A027187, A028260, A239830, A341446, A344609, A344616, A344651, A345197, A345910, A345912, A345961.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[primeMS[#]]==-2&]

A191529 Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) with no initial and no final (1,0)-steps.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 6, 5, 20, 21, 70, 84, 252, 330, 924, 1287, 3432, 5005, 12870, 19448, 48620, 75582, 184756, 293930, 705432, 1144066, 2704156, 4457400, 10400600, 17383860, 40116600, 67863915, 155117520, 265182525, 601080390, 1037158320, 2333606220, 4059928950, 9075135300

Offset: 0

Emeric Deutsch, Jun 07 2011

			a(6)=6 because we have UDHHUD and the 5 Dyck paths of length 6: UDUDUD, UDUUDD, UUDDUD, UUDUDD, and UUUDDD; here U=(1,1), H=(1,0) and D=(1,-1).

Cf. A000984, A002054.

a := proc (n) if `mod`(n, 2) = 0 then binomial(n-2, (1/2)*n-1) else binomial(n-2, (1/2)*n-5/2) end if end proc: 1, 0, seq(a(n), n = 2 .. 38);
A191529:=n->binomial(n,floor(n/2)) - 2*binomial(n-1,floor((n-1)/2)) + binomial(n-2,floor((n-2)/2)) + 2*0^n: seq(A191529(n), n=0..40); # Wesley Ivan Hurt, Sep 27 2014

Join[{1}, Table[Binomial[n, Floor[n/2]] - 2 Binomial[n - 1, Floor[(n - 1)/2]] + Binomial[n - 2, Floor[(n - 2)/2]], {n, 40}]] (* Wesley Ivan Hurt, Sep 27 2014 *)

a(2n) = binomial(2n-2,n-1) = A000984(n-1) (n>=1).
a(2n+1) = binomial(2n-1,n-2) = A002054(n-1) (n>=1).
G.f.: g(z)=1+(1-z)(1-q)/(1-2z+q), where q=sqrt(1-4z^2).
a(n) = binomial(n,floor(n/2)) - 2*binomial(n-1,floor((n-1)/2)) + binomial(n-2,floor((n-2)/2)) + 2*0^n. - Wesley Ivan Hurt, Sep 27 2014
D-finite with recurrence (n+1)*a(n) -n*a(n-1) +2*(-2*n+3)*a(n-2) +4*(n-4)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

cp's OEIS Frontend

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A146305 Array T(n,m) = 2(2m+3)!(4n+2m+1)!/(m!(m+2)!n!*(3n+2m+3)!) read by antidiagonals.