cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A287847 Number A(n,k) of Dyck paths of semilength n such that no level has more than k peaks; square array A(n,k), n >= 0, k >= 0, read by descending antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 3, 0, 1, 1, 2, 4, 5, 0, 1, 1, 2, 5, 12, 13, 0, 1, 1, 2, 5, 13, 31, 31, 0, 1, 1, 2, 5, 14, 40, 90, 71, 0, 1, 1, 2, 5, 14, 41, 119, 264, 181, 0, 1, 1, 2, 5, 14, 42, 130, 376, 797, 447, 0, 1, 1, 2, 5, 14, 42, 131, 414, 1202, 2402, 1111, 0
Offset: 0

Views

Author

Alois P. Heinz, Jun 01 2017

Keywords

Examples

			. A(3,1) = 3:                    /\
.                 /\    /\      /  \
.              /\/  \  /  \/\  /    \   .
.
. A(3,2) = 4:                            /\
.                 /\    /\      /\/\    /  \
.              /\/  \  /  \/\  /    \  /    \   .
.
. A(3,3) = 5:                                    /\
.                         /\    /\      /\/\    /  \
.              /\/\/\  /\/  \  /  \/\  /    \  /    \   .
.
Square array A(n,k) begins:
  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,   2,   2,   2,   2,   2,   2, ...
  0,  3,   4,   5,   5,   5,   5,   5, ...
  0,  5,  12,  13,  14,  14,  14,  14, ...
  0, 13,  31,  40,  41,  42,  42,  42, ...
  0, 31,  90, 119, 130, 131, 132, 132, ...
  0, 71, 264, 376, 414, 427, 428, 429, ...

Links

Crossrefs

Columns k=0-10 give: A000007, A281874, A287966, A287967, A287968, A287969, A287970, A287971, A287972, A287973, A287974.
Main diagonal and first two lower diagonals give: A000108, A001453, A120304.
Cf. A287822.

Programs

  • Maple
    b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1, (m->
      add(b(n, m, j), j=1..m))(min(n, k)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
  • Mathematica
    b[n_, k_, j_] := b[n, k, j] = If[j == n, 1, Sum[b[n - j, k, i]*Sum[ Binomial[i, m]*Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, 1, Min[j + k, n - j]}]];
A[n_, k_] := A[n, k] = If[n==0, 1, Sum[b[n, #, j], {j, 1, #}]&[Min[n, k]]];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum([b(n - j, k, i)*sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)]) for i in range(1, min(j + k, n - j) + 1)])
@cacheit
def A(n, k):
    if n==0: return 1
    m=min(n, k)
    return sum([b(n, m , j) for j in range(1, m + 1)])
for d in range(21): print([A(n, d - n) for n in range(d + 1)]) # Indranil Ghosh, Aug 16 2017

Formula

A(n,k) = Sum_{j=0..k} A287822(n,j).

A288387 Number T(n,k) of Dyck paths of semilength n such that the minimal number of peaks over all positive levels equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 8, 5, 0, 0, 1, 25, 13, 3, 0, 0, 1, 83, 35, 13, 0, 0, 0, 1, 282, 112, 30, 4, 0, 0, 0, 1, 971, 368, 61, 29, 0, 0, 0, 0, 1, 3386, 1208, 172, 90, 5, 0, 0, 0, 0, 1, 11940, 3992, 619, 188, 56, 0, 0, 0, 0, 0, 1, 42504, 13449, 2241, 345, 240, 6, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2017

Keywords

Comments

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms for k<=n. T(n,k) = 0 if k>n.
T(0,0) = 1 by convention.

Examples

			. T(4,1) = 5:
.              /\      /\        /\/\    /\        /\/\
.         /\/\/  \  /\/  \/\  /\/    \  /  \/\/\  /    \/\ .
.
Triangle T(n,k) begins:
:    1;
:    0,    1;
:    1,    0,   1;
:    2,    2,   0,  1;
:    8,    5,   0,  0, 1;
:   25,   13,   3,  0, 0, 1;
:   83,   35,  13,  0, 0, 0, 1;
:  282,  112,  30,  4, 0, 0, 0, 1;
:  971,  368,  61, 29, 0, 0, 0, 0, 1;
: 3386, 1208, 172, 90, 5, 0, 0, 0, 0, 1;

Links

Crossrefs

Columns k=0-10 give: A288539, A288540, A288541, A288542, A288543, A288544, A288545, A288546, A288547, A288548, A288549.
Row sums give A000108.
Main diagonal and first lower diagonal give: A000012, A000004.
Cf. A000007, A000027, A218152, A287822, A288386.

Programs

  • Maple
    b:= proc(n, k, j) option remember; `if`(j=n, 1,
      add(add(binomial(i, m)*binomial(j-1, i-1-m),
      m=max(k, i-j)..i-1)*b(n-j, k, i), i=1..n-j))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n, k, j), j=k..n))
    end:
T:= (n, k)-> `if`(n=k, 1, A(n, k)-A(n, k+1)):
seq(seq(T(n, k), k=0..n), n=0..14);
  • Mathematica
    b[n_, k_, j_] := b[n, k, j] = If[j==n, 1, Sum[Sum[Binomial[i, m]*Binomial[ j-1, i-1-m], {m, Max[k, i - j], i - 1}]*b[n - j, k, i], {i, 1, n - j}]];
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[b[n, k, j], {j, k, n}]];
T[n_, k_] := If[n == k, 1, A[n, k] - A[n, k + 1]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)

Formula

T(0,0) = 1, T(n,k) = A288386(n,k) - A288386(n,k+1).
T(2n,n-1) = A218152(n) for n>1.
T(2n,n) = A000007(n).
T(2n+1,n) = A000027(n+1) for n>0.

A287860 Number of Dyck paths of semilength 2n such that the maximal number of peaks per level equals n.

Original entry on oeis.org

1, 1, 7, 29, 163, 925, 5580, 34751, 222627, 1456952, 9699872, 65474460, 446971110, 3080074508, 21393773841, 149614083615, 1052537452164, 7443584137525, 52888757972865, 377382278671610, 2703141489113003, 19430405608302831, 140118758417377105
Offset: 0

Views

Author

Alois P. Heinz, Jun 01 2017

Keywords

Examples

			.                  /\       /\         /\/\
.  a(2) = 7:  /\/\/  \   /\/  \/\   /\/    \
.
.                                     /\/\
.   /\         /\  /\     /\/\       /    \
.  /  \/\/\   /  \/  \   /    \/\   /      \  .

Links

Crossrefs

Cf. A287822.

Programs

  • Maple
    b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> `if`(n=0, 1, g(2*n, n)-g(2*n, n-1)):
seq(a(n), n=0..23);
  • Mathematica
    b[n_, k_, j_] := b[n, k, j] = If[j == n, 1, Sum[b[n - j, k, i]*Sum[ Binomial[i, m] * Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, 1, Min[j + k, n - j]}]];
g[n_, k_] := g[n, k] = Sum[b[n, k, j], {j, 1, k}];
a[n_] := If[n == 0, 1, g[2*n, n] - g[2*n, n - 1]];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Formula

a(n) = A287822(2n,n).

A288743 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals two.

Original entry on oeis.org

1, 1, 7, 18, 59, 193, 616, 1955, 6244, 19926, 63490, 202068, 642816, 2044571, 6502193, 20673020, 65714586, 208870774, 663868055, 2109997964, 6706282384, 21315049217, 67748772174, 215343287489, 684507346839, 2175916952697, 6917096914771, 21989855308501
Offset: 2

Views

Author

Alois P. Heinz, Jun 14 2017

Keywords

Examples

			. a(4) = 7:       /\      /\        /\/\    /\        /\  /\
.            /\/\/  \  /\/  \/\  /\/    \  /  \/\/\  /  \/  \ .
.
.                        /\/\
.             /\/\      /    \
.            /    \/\  /      \  .

Links

Crossrefs

Column k=2 of A287822.
Cf. A000108.

Programs

  • Maple
    b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 2)-g(n, 1):
seq(a(n), n=2..35);
  • Mathematica
    b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j, k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 2] - g[n, 1], {n, 2, 35}] (* Indranil Ghosh, Aug 09 2017 *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum(b(n - j, k, i)*sum(binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)) for i in range(1, min(j + k, n - j) + 1))
def g(n, k): return sum(b(n, k, j) for j in range(1, k + 1))
def a(n): return g(n, 2) - g(n, 1)
print([a(n) for n in range(2, 36)]) # Indranil Ghosh, Aug 09 2017

A288744 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals three.

Original entry on oeis.org

1, 1, 9, 29, 112, 405, 1514, 5565, 20249, 73416, 265616, 957677, 3441282, 12329838, 44062706, 157105923, 559009643, 1985301783, 7038496811, 24913917722, 88058727525, 310832221932, 1095854282575, 3859201682187, 13576884290502, 47719628447310, 167579774234059
Offset: 3

Views

Author

Alois P. Heinz, Jun 14 2017

Keywords

Examples

			. T(4) = 1:   /\/\/\
.            /      \  .

Links

Crossrefs

Column k=3 of A287822.
Cf. A000108.

Programs

  • Maple
    b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 3)-g(n, 2):
seq(a(n), n=3..35);
  • Mathematica
    b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j, k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 3] - g[n, 2], {n, 3, 35}] (* Indranil Ghosh, Aug 09 2017 *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum(b(n - j, k, i)*sum(binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)) for i in range(1, min(j + k, n - j) + 1))
def g(n, k): return sum(b(n, k, j) for j in range(1, k + 1))
def a(n): return g(n, 3) - g(n, 2)
print([a(n) for n in range(3, 36)]) # Indranil Ghosh, Aug 09 2017

A288745 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals four.

Original entry on oeis.org

1, 1, 11, 38, 163, 648, 2571, 10173, 40025, 156087, 605057, 2335566, 8980883, 34412583, 131431024, 500437733, 1900135511, 7196366668, 27191450135, 102522926104, 385785153584, 1448985664032, 5432879981201, 20337296148823, 76015000686028, 283720418696600
Offset: 4

Views

Author

Alois P. Heinz, Jun 14 2017

Keywords

Links

Crossrefs

Column k=4 of A287822.
Cf. A000108.

Programs

  • Maple
    b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 4)-g(n, 3):
seq(a(n), n=4..35);
  • Mathematica
    b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j,k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 4] - g[n, 3], {n, 4, 35}] (* Indranil Ghosh, Aug 08 2017 *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum([b(n - j, k, i)*sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)]) for i in range(1, min(j + k, n - j) + 1)])
def g(n, k): return sum([b(n, k, j) for j in range(1, k + 1)])
def a(n): return g(n, 4) - g(n, 3)
print([a(n) for n in range(4, 36)]) # Indranil Ghosh, Aug 08 2017

A288746 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals five.

Original entry on oeis.org

1, 1, 13, 48, 220, 925, 3895, 16137, 66399, 271446, 1101626, 4442143, 17822176, 71191082, 283269813, 1123212251, 4439583152, 17496345670, 68765995160, 269595218881, 1054499461385, 4115767918639, 16032123369549, 62333852291879, 241935803355457, 937486479689517
Offset: 5

Views

Author

Alois P. Heinz, Jun 14 2017

Keywords

Links

Crossrefs

Column k=5 of A287822.
Cf. A000108.

Programs

  • Maple
    b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 5)-g(n, 4):
seq(a(n), n=5..35);
  • Mathematica
    b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j,k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 5] - g[n, 4], {n, 5, 35}] (* Indranil Ghosh, Aug 08 2017 *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum([b(n - j, k, i)*sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)]) for i in range(1, min(j + k, n - j) + 1)])
def g(n, k): return sum([b(n, k, j) for j in range(1, k + 1)])
def a(n): return g(n, 5) - g(n, 4)
print([a(n) for n in range(5, 36)]) # Indranil Ghosh, Aug 08 2017

A288747 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals six.

Original entry on oeis.org

1, 1, 15, 59, 288, 1269, 5580, 24092, 102847, 434794, 1824249, 7600076, 31459191, 129505739, 530589496, 2164696038, 8798355232, 35639564649, 143919521948, 579526079335, 2327484839124, 9324921648372, 37275509745894, 148692946409186, 591979810055622
Offset: 6

Views

Author

Alois P. Heinz, Jun 14 2017

Keywords

Links

Crossrefs

Column k=6 of A287822.
Cf. A000108.

Programs

  • Maple
    b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 6)-g(n, 5):
seq(a(n), n=6..35);
  • Mathematica
    b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j,k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 6] - g[n, 5], {n, 6, 35}] (* Indranil Ghosh, Aug 08 2017 *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum([b(n - j, k, i)*sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)]) for i in range(1, min(j + k, n - j) + 1)])
def g(n, k): return sum([b(n, k, j) for j in range(1, k + 1)])
def a(n): return g(n, 6) - g(n, 5)
print([a(n) for n in range(6, 36)]) # Indranil Ghosh, Aug 08 2017

A288748 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals seven.

Original entry on oeis.org

1, 1, 17, 71, 368, 1697, 7769, 34751, 153313, 668088, 2882104, 12329145, 52358300, 220901081, 926638057, 3867432363, 16068748557, 66495876593, 274178902925, 1126793986670, 4616878543095, 18864740697016, 76885237242318, 312611605360287, 1268261191750753
Offset: 7

Views

Author

Alois P. Heinz, Jun 14 2017

Keywords

Links

Crossrefs

Column k=7 of A287822.
Cf. A000108.

Programs

  • Maple
    b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 7)-g(n, 6):
seq(a(n), n=7..35);
  • Mathematica
    b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j,k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 7] - g[n, 6], {n, 7, 35}] (* Indranil Ghosh, Aug 08 2017 *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum([b(n - j, k, i)*sum([binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)]) for i in range(1, min(j + k, n - j) + 1)])
def g(n, k): return sum([b(n, k, j) for j in range(1, k + 1)])
def a(n): return g(n, 7) - g(n, 6)
print([a(n) for n in range(7, 36)]) # Indranil Ghosh, Aug 08 2017

A288749 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals eight.

Original entry on oeis.org

1, 1, 19, 84, 461, 2222, 10577, 48943, 222627, 997735, 4417674, 19359659, 84099436, 362570722, 1552681071, 6609823112, 27989970166, 117967914457, 495087382572, 2069827499508, 8623283249034, 35811917284318, 148289870077879, 612382134256433, 2522591250558641
Offset: 8

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Author

Alois P. Heinz, Jun 14 2017

Keywords

Links

Crossrefs

Column k=8 of A287822.
Cf. A000108.

Programs

  • Maple
    b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 8)-g(n, 7):
seq(a(n), n=8..35);
  • Mathematica
    b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j,k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 8] - g[n, 7], {n, 8, 35}] (* Indranil Ghosh, Aug 08 2017 *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum(b(n - j, k, i)*sum(binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)) for i in range(1, min(j + k, n - j) + 1))
def g(n, k): return sum(b(n, k, j) for j in range(1, k + 1))
def a(n): return g(n, 8) - g(n, 7)
print([a(n) for n in range(8, 36)]) # Indranil Ghosh, Aug 08 2017
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