A014413 -id:A014413 - cp's OEIS frontend

A034868 Left half of Pascal's triangle.

1, 1, 1, 2, 1, 3, 1, 4, 6, 1, 5, 10, 1, 6, 15, 20, 1, 7, 21, 35, 1, 8, 28, 56, 70, 1, 9, 36, 84, 126, 1, 10, 45, 120, 210, 252, 1, 11, 55, 165, 330, 462, 1, 12, 66, 220, 495, 792, 924, 1, 13, 78, 286, 715, 1287, 1716, 1, 14, 91, 364, 1001, 2002, 3003, 3432, 1, 15

Offset: 0

N. J. A. Sloane

			1;
1;
1, 2;
1, 3;
1, 4,  6;
1, 5, 10;
1, 6, 15, 20;
...

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A107430, A062344, A122366, A027306 (row sums).
Cf. A008619.
Cf. A225860.
Cf. A126257.
Cf. A034869 (right half), A014413, A014462, A265848.

a034868 n k = a034868_tabf !! n !! k
a034868_row n = a034868_tabf !! n
a034868_tabf = map reverse a034869_tabf
-- Reinhard Zumkeller, improved Dec 20 2015, Jul 27 2012

Flatten[ Table[ Binomial[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}]] (* Robert G. Wilson v, May 28 2005 *)

for(n=0, 14, for(k=0, floor(n/2), print1(binomial(n, k),", ");); print();) \\ Indranil Ghosh, Mar 31 2017

import math
from sympy import binomial
for n in range(15):
    print([binomial(n, k) for k in range(int(math.floor(n/2)) + 1)]) # Indranil Ghosh, Mar 31 2017

from itertools import count, islice
def A034868_gen(): # generator of terms
    yield from (s:=(1,))
    for i in count(0):
        yield from (s:=(1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1)) + ((s[-1]<<1,) if i&1 else ()))
A034868_list = list(islice(A034868_gen(),30)) # Chai Wah Wu, Oct 17 2023

T(n,k) = A034869(n,floor(n/2)-k), k = 0..floor(n/2). - Reinhard Zumkeller, Jul 27 2012

A034869 Right half of Pascal's triangle.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 6, 4, 1, 10, 5, 1, 20, 15, 6, 1, 35, 21, 7, 1, 70, 56, 28, 8, 1, 126, 84, 36, 9, 1, 252, 210, 120, 45, 10, 1, 462, 330, 165, 55, 11, 1, 924, 792, 495, 220, 66, 12, 1, 1716, 1287, 715, 286, 78, 13, 1, 3432, 3003, 2002, 1001, 364, 91, 14, 1

Offset: 0

N. J. A. Sloane

From R. J. Mathar, May 13 2006: (Start)
Also flattened table of the expansion coefficients of x^n in Chebyshev Polynomials T_k(x) of the first kind:
x^n is 2^(1-n) multiplied by the sum of floor(1+n/2) terms using only terms T_k(x) with even k if n even, only terms T_k(x) with odd k if n is odd and halving the coefficient a(..) in front of any T_0(x):
x^0=2^(1-0) a(0)/2 T_0(x)
x^1=2^(1-1) a(1) T_1(x)
x^2=2^(1-2) [a(2)/2 T_0(x)+a(3) T_2(x)]
x^3=2^(1-3) [a(4) T_1(x)+a(5) T_3(x)]
x^4=2^(1-4) [a(6)/2 T_0(x)+a(7) T_2(x) +a(8) T_4(x)]
x^5=2^(1-5) [a(9) T_1(x)+a(10) T_3(x) +a(11) T_5(x)]
x^6=2^(1-6) [a(12)/2 T_0(x)+a(13) T_2(x) +a(14) T_4(x) +a(15) T_6(x)]
x^7=2^(1-7) [a(16) T_1(x)+a(17) T_3(x) +a(18) T_5(x) +a(19) T_7(x)]" (End)
T(n,k) = A034868(n,floor(n/2)-k), k = 0..floor(n/2). - Reinhard Zumkeller, Jul 27 2012
Rows are binomial(r-1,(2r+1-(-1)^r)\4 -n ) where r is the row and n is the term. Columns are binomial(2m+c-3,m-1) where c is the column and m is the term. - Anthony Browne, May 17 2016

			The table starts:
  1
  1
  2 1
  3 1
  6 4 1
  ...

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Cf. A007318, A008619 (row lengths).
Cf. A110654.
Cf. A034868 (left half), A014413, A014462, A027306 (row sums).
Columns k=0-1-2-3-4 give: A001405, A037955, A037956, A037957, A037958.

a034869 n k = a034869_tabf !! n !! k
a034869_row n = a034869_tabf !! n
a034869_tabf = [1] : f 0 [1] where
   f 0 us'@(_:us) = ys : f 1 ys where
                    ys = zipWith (+) us' (us ++ [0])
   f 1 vs@(v:_) = ys : f 0 ys where
                  ys = zipWith (+) (vs ++ [0]) ([v] ++ vs)
-- Reinhard Zumkeller, improved Dec 21 2015, Jul 27 2012

for n from 0 to 60 do for j from n mod 2 to n by 2 do print( binomial(n,(n-j)/2) ); od; od; # R. J. Mathar, May 13 2006
# Second program:
egf:= k-> BesselI(2*k, 2*x) + BesselI(2*k+1, 2*x):
A034869:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A034869(n, k), k=0..iquo(n, 2))), n=0..14); # Mélika Tebni, Sep 05 2024

Table[Binomial[n, k], {n, 0, 14}, {k, Ceiling[n/2], n}] // Flatten (* Michael De Vlieger, May 19 2016 *)

for(n=0, 14, for(k=ceil(n/2), n, print1(binomial(n, k),", ");); print();) \\ Indranil Ghosh, Mar 31 2017

import math
from sympy import binomial
for n in range(15):
    print([binomial(n, k) for k in range(math.ceil(n/2), n + 1)]) # Indranil Ghosh, Mar 31 2017

E.g.f. of column k: BesselI(2*k,2*x) + BesselI(2*k+1,2*x). - Mélika Tebni, Sep 05 2024

Keyword fixed and example added by Franklin T. Adams-Watters, May 27 2010

A014462 Triangular array formed from elements to left of middle of Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 5, 10, 1, 6, 15, 1, 7, 21, 35, 1, 8, 28, 56, 1, 9, 36, 84, 126, 1, 10, 45, 120, 210, 1, 11, 55, 165, 330, 462, 1, 12, 66, 220, 495, 792, 1, 13, 78, 286, 715, 1287, 1716, 1, 14, 91, 364, 1001, 2002, 3003, 1, 15, 105, 455, 1365, 3003, 5005, 6435, 1, 16

Offset: 1

Mohammad K. Azarian

Coefficients for Pontryagin classes of projective spaces. See p. 3 of the Wilson link. Aerated to become a lower triangular matrix with alternating zeros on the diagonal, this matrix appparently becomes the reverse, or mirror, of A117178. - Tom Copeland, May 30 2017

			Array begins:
  1;
  1;
  1,  3;
  1,  4;
  1,  5, 10;
  1,  6, 15;
  1,  7, 21,  35;
  1,  8, 28,  56;
  1,  9, 36,  84, 126;
  1, 10, 45, 120, 210;
  1, 11, 55, 165, 330, 462;

Reinhard Zumkeller, Rows n = 1..200 of triangle, flattened
Jon Maiga, Computer-generated formulas for A014462, Sequence Machine.
D. Wilson, Genera: From cobordism to K-theory, Jul 2016.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A014413, A034868, A058622 (row sums).
Cf. A001791 (a half-diagonal and diagonal sums).
Cf. A007318, A000037, A008314.
Cf. A117178.

a014462 n k = a014462_tabf !! (n-1) !! (k-1)
a014462_row n = a014462_tabf !! (n-1)
a014462_tabf = map reverse a014413_tabf
-- Reinhard Zumkeller, Dec 24 2015

More terms from James Sellers

A028330 Elements to the right of the central elements of the even-Pascal triangle A028326.

Original entry on oeis.org

2, 2, 6, 2, 8, 2, 20, 10, 2, 30, 12, 2, 70, 42, 14, 2, 112, 56, 16, 2, 252, 168, 72, 18, 2, 420, 240, 90, 20, 2, 924, 660, 330, 110, 22, 2, 1584, 990, 440, 132, 24, 2, 3432, 2574, 1430, 572, 156, 26, 2, 6006, 4004, 2002, 728, 182, 28, 2, 12870, 10010, 6006, 2730

Offset: 0

Mohammad K. Azarian

			This sequence represents the following portion of A028330(n,k), with x being the elements of A028329(n):
  x;
  .,  2;
  .,  x,  2;
  .,  .,  6,  2;
  .,  .,  x,  8,  2;
  .,  .,  ., 20, 10,   2;
  .,  .,  .,  x, 30,  12,   2;
  .,  .,  .,  ., 70,  42,  14,    2;
  .,  .,  .,  .,  x, 112,  56,   16,   2;
  .,  .,  .,  .,  ., 252, 168,   72,  18,   2;
  .,  .,  .,  .,  .,   x, 420,  240,  90,  20,   2;
  .,  .,  .,  .,  .,   ., 924,  660, 330, 110,  22,  2;
  .,  .,  .,  .,  .,   .,   x, 1584, 990, 440, 132, 24, 2;
As an irregular triangle:
    2;
    2;
    6,   2;
    8,   2;
   20,  10,   2;
   30,  12,   2;
   70,  42,  14,   2;
  112,  56,  16,   2;
  252, 168,  72,  18,  2;
  420, 240,  90,  20,  2;
  924, 660, 330, 110, 22,  2;

G. C. Greubel, Table of n, a(n) for n = 0..2549

Cf. A028326, A028327, A028328, A028329, A028331, A028332.

Cf. A001405, A014413, A058622, A063886, A202736.

[[2*Binomial(n,k): k in [Floor((n+2)/2)..n]]: n in [1..12]]; // G. C. Greubel, Jul 14 2024

Table[2*Binomial[n+1, k+1 +Floor[(n+1)/2]], {n,0,12}, {k,0,Floor[n/2] }]//Flatten (* G. C. Greubel, Jul 14 2024 *)

def A028326(n,k): return 2*binomial(n, k)
flatten([[A028326(n,k) for k in range(((n+2)//2), n+1)] for n in range(1,21)]) # G. C. Greubel, Jul 14 2024

a(n) = 2 * A014413(n). - Sean A. Irvine, Dec 29 2019
From G. C. Greubel, Jul 14 2024: (Start)
T(n, k) = 2*binomial(n+1, k+1 + floor((n+1)/2)) for n >= 0, 0 <= k <= floor(n/2).
Sum_{k=0..floor(n/2)} T(n, k) = A202736(n+1) = 2*A058622(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n, k) = 2*A001405(n) = A063886(n+1). (End)

More terms from James Sellers

A265848 Pascal's triangle, right and left halves interchanged.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Dec 24 2015

Concatenations of rows of A014413 and A034868.

Alternative mirrored variant: concatenation of A034869 and A014462.

			.   0:                                1
.   1:                              1   1
.   2:                            1   1   2
.   3:                          3   1   1   3
.   4:                        4   1   1   4   6
.   5:                     10   5   1   1   5   10
.   6:                   15   6   1   1   6   15  20
.   7:                 35   21  7   1   1   7   21   35
.   8:              56   28   8   1   1   8   28  56   70
.   9:           126   84   36  9   1   1   9   36   84   126
.  10:        210   120  45  10   1   1   10  45  120  210  252
.  11:     462   330  165   55  11  1   1   11  55  165   330  462
.  12:  792   495  220   66  12   1   1   12  66  220  495  792   924  .

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle.

Cf. A014413, A014462, A034868, A034869, A007318, A001405, A037952, A000079 (row sums), A001142 (row products).

a265848 n k = a265848_tabl !! n !! k
a265848_row n = a265848_tabl !! n
a265848_tabl = zipWith (++) ([] : a014413_tabf) a034868_tabf

row[n_] := Binomial[n, Join[Range[Floor[n/2] + 1, n], Range[0, Floor[n/2]]]]; Array[row, 12, 0] // Flatten (* Amiram Eldar, May 13 2025 *)

T(n,k) = A007318(n, (k + floor((n+2)/2)) mod (n+1)).
T(n,k) = if k <= [(n+1)/2] then A014413(n,k+1) else A034868(n,k-[(n+1)/2]).
T(n,0) = A037952(n) for n > 0.
T(n,n) = A001405(n).

A014721 Squares of elements to left of the central element in Pascal triangle (by row).

Original entry on oeis.org

1, 1, 9, 1, 16, 1, 100, 25, 1, 225, 36, 1, 1225, 441, 49, 1, 3136, 784, 64, 1, 15876, 7056, 1296, 81, 1, 44100, 14400, 2025, 100, 1, 213444, 108900, 27225, 3025, 121, 1, 627264, 245025, 48400, 4356, 144, 1, 2944656, 1656369, 511225, 81796

Offset: 1

Mohammad K. Azarian

Cf. A014413.

a(n) = A014413(n)^2. - Sean A. Irvine, Nov 18 2018

More terms from Sean A. Irvine, Nov 18 2018

Offset corrected by Joerg Arndt, Nov 19 2018

cp's OEIS Frontend

A034868 Left half of Pascal's triangle.

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A034869 Right half of Pascal's triangle.

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A014462 Triangular array formed from elements to left of middle of Pascal's triangle.

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A028330 Elements to the right of the central elements of the even-Pascal triangle A028326.

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A265848 Pascal's triangle, right and left halves interchanged.

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A014721 Squares of elements to left of the central element in Pascal triangle (by row).

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