A109449 -id:A109449 - cp's OEIS frontend

A247453 T(n,k) = binomial(n,k)A000111(n-k)(-1)^(n-k), 0 <= k <= n.

1, -1, 1, 1, -2, 1, -2, 3, -3, 1, 5, -8, 6, -4, 1, -16, 25, -20, 10, -5, 1, 61, -96, 75, -40, 15, -6, 1, -272, 427, -336, 175, -70, 21, -7, 1, 1385, -2176, 1708, -896, 350, -112, 28, -8, 1, -7936, 12465, -9792, 5124, -2016, 630, -168, 36, -9, 1, 50521

Offset: 0

Reinhard Zumkeller, Sep 17 2014

Matrix inverse of A109449, the unsigned version of this sequence. More precisely, consider both of these triangles as the nonzero lower left of an infinite square array / matrix, filled with zeros above/right of the diagonal. Then these are mutually inverse of each other; in matrix notation: A247453 . A109449 = A109449 . A247453 = Identity matrix. In more conventional notation, for any m,n >= 0, Sum_{k=0..n} A247453(n,k)*A109449(k,m) = Sum_{k=0..n} A109449(n,k)*A247453(k,m) = delta(m,n), the Kronecker delta (= 1 if m = n, 0 else). - M. F. Hasler, Oct 06 2017

			.   0:      1
.   1:     -1      1
.   2:      1     -2      1
.   3:     -2      3     -3      1
.   4:      5     -8      6     -4      1
.   5:    -16     25    -20     10     -5     1
.   6:     61    -96     75    -40     15    -6     1
.   7:   -272    427   -336    175    -70    21    -7    1
.   8:   1385  -2176   1708   -896    350  -112    28   -8   1
.   9:  -7936  12465  -9792   5124  -2016   630  -168   36  -9   1
.  10:  50521 -79360  62325 -32640  12810 -4032  1050 -240  45 -10  1  .

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
OEIS Wiki, Boustrophedon transform.
Wikipedia, Boustrophedon transform
Index entries for sequences related to boustrophedon transform

Cf. A000111, A007318, A062162, A109449.

a247453 n k = a247453_tabl !! n !! k
a247453_row n = a247453_tabl !! n
a247453_tabl = zipWith (zipWith (*)) a109449_tabl a097807_tabl

a111[n_] := n! SeriesCoefficient[(1+Sin[x])/Cos[x], {x, 0, n}];
T[n_, k_] := (-1)^(n-k) Binomial[n, k] a111[n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 03 2018 *)

A247453(n,k)=(-1)^(n-k)*binomial(n,k)*if(n>k, 2*abs(polylog(k-n, I)), 1) \\ M. F. Hasler, Oct 06 2017

T(n,k) = (-1)^(n-k) * A007318(n,k) * A000111(n-k), k = 0..n;
T(n,k) = (-1)^(n-k) * A109449(n,k); A109449(n,k) = abs(T(n,k));
abs(sum of row n) = A062162(n);
Sum_{k=0..n} T(n,k)*A000111(k) = 0^n.

Edited by M. F. Hasler, Oct 06 2017

A000718 Boustrophedon transform of triangular numbers 1,1,3,6,10,...

Original entry on oeis.org

1, 2, 6, 20, 65, 226, 883, 3947, 20089, 115036, 732171, 5126901, 39165917, 324138010, 2888934623, 27587288507, 281001801969, 3041152133848, 34849036364659, 421526126267265, 5367037330561365, 71752003756908550

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A109449, A000217, A000746.

a000718 n = sum $ zipWith (*) (a109449_row n) (1 : tail a000217_list)
-- Reinhard Zumkeller, Nov 04 2013

t[n_, 0] := If[n == 0, 1, n*(n+1)/2]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import accumulate, count, islice
def A000718_gen(): # generator of terms
    yield 1
    blist, c = (1,), 1
    for i in count(2):
        yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
        c += i
A000718_list = list(islice(A000718_gen(),30)) # Chai Wah Wu, Jun 11 2022

E.g.f.: (sec(x) + tan(x))*(exp(x)*(x + 1/2*x^2) + 1). - Sergei N. Gladkovskii, Oct 30 2014

a(n) ~ n! * (8 + exp(Pi/2)*Pi*(4+Pi)) * 2^(n-1) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

A000732 Boustrophedon transform of 1 & primes: 1,2,3,5,7,...

Original entry on oeis.org

1, 3, 8, 22, 66, 222, 862, 3838, 19542, 111894, 712282, 4987672, 38102844, 315339898, 2810523166, 26838510154, 273374835624, 2958608945772, 33903161435148, 410085034127000, 5221364826476796, 69804505809732988

Offset: 0

N. J. A. Sloane and Simon Plouffe

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform.
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A000747, A230953, A230954, A230955.

a000732 n = sum $ zipWith (*) (a109449_row n) a008578_list

t[n_, 0] := If[n==0, 1, Prime[n]]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import accumulate, count, islice
from sympy import prime
def A000732_gen(): # generator of terms
    yield 1
    blist = (1,)
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=prime(i))))[-1]
A000732_list = list(islice(A000732_gen(),40)) # Chai Wah Wu, Jun 12 2022

a(n) = Sum_{k=0..n} A109449(n,k)*A008578(k+1). - Reinhard Zumkeller, Nov 04 2013

E.g.f.: (sec(x) + tan(x))*(1 + Sum_{k>=1} prime(k)*x^k/k!). - Ilya Gutkovskiy, Apr 23 2019

A000734 Boustrophedon transform of 1,1,2,4,8,16,32,...

Original entry on oeis.org

1, 2, 5, 15, 49, 177, 715, 3255, 16689, 95777, 609875, 4270695, 32624329, 269995377, 2406363835, 22979029335, 234062319969, 2533147494977, 29027730898595, 351112918079175, 4470508510495609, 59766296291090577

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

Binomial transform of A062272. - Paul Barry, Jan 21 2005

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A109449, A000079, A000752.

a000734 n = sum $ zipWith (*) (a109449_row n) (1 : a000079_list)
-- Reinhard Zumkeller, Nov 04 2013

CoefficientList[Series[(1+E^(2*x))*(Sec[x]+Tan[x])/2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 07 2013 *)
t[n_, 0] := If[n == 0, 1, 2^(n-1)]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n - 1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import count, accumulate, islice
def A000734_gen(): # generator of terms
    yield 1
    blist, m = (1,), 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m *= 2
A000734_list = list(islice(A000734_gen(),40)) # Chai Wah Wu, Jun 12 2022

# Algorithm of L. Seidel (1877)
def A000734_list(n) :
    A = {-1:0, 0:1}; R = []
    k = 0; e = 1; Bm = 1
    for i in range(n) :
        Am = Bm
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        Bm += Bm
        R.append(A[e*i//2]/2)
    return R
A000734_list(22) # Peter Luschny, Jun 02 2012

E.g.f.: (1 + exp(2*x))*(sec(x) + tan(x))/2. - Paul Barry, Jan 21 2005

a(n) ~ n! * (1 + exp(Pi)) * (2/Pi)^(n+1). - Vaclav Kotesovec, Oct 07 2013

A000751 Boustrophedon transform of partition numbers.

Original entry on oeis.org

1, 2, 5, 14, 42, 143, 555, 2485, 12649, 72463, 461207, 3229622, 24671899, 204185616, 1819837153, 17378165240, 177012514388, 1915724368181, 21952583954117, 265533531724484, 3380877926676504, 45199008472762756

Offset: 0

N. J. A. Sloane

nonn

			The array begins:
                   1
               1  ->   2
           5  <-   4  <-   2
       3  ->   8  ->  12  ->  14
  42  <-  39  <-  31  <-  19  <-   5
- _John Cerkan_, Jan 26 2017

John Cerkan, Table of n, a(n) for n = 0..482
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms
Wikipedia, Boustrophedon_transform
Index entries for sequences related to boustrophedon transform

Cf. A000733, A230957.

a000751 n = sum $ zipWith (*) (a109449_row n) a000041_list
-- Reinhard Zumkeller, Nov 03 2013

t[n_, 0] := PartitionsP[n]; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import accumulate, count, islice
from sympy import npartitions
def A000751_gen(): # generator of terms
    blist = tuple()
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist),initial=npartitions(i))))[-1]
A000751_list = list(islice(A000751_gen(),40)) # Chai Wah Wu, Jun 12 2022

a(n) = Sum_{k=0..n} A109449(n,k)*A000041(k). - Reinhard Zumkeller, Nov 03 2013

A000754 Boustrophedon transform of odd numbers.

Original entry on oeis.org

1, 4, 12, 33, 96, 317, 1218, 5425, 27608, 158129, 1006574, 7048657, 53847420, 445643681, 3971876930, 37928628529, 386337833232, 4181155148673, 47912508680086, 579538956964241, 7378919177090244, 98648882783190305

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform.
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996), 44-54 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A005408.

a000754 n = sum $ zipWith (*) (a109449_row n) [1, 3 ..]
-- Reinhard Zumkeller, Nov 02 2013

CoefficientList[Series[(Sec[x]+Tan[x])*E^x*(2*x+1), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 30 2014 after Sergei N. Gladkovskii *)
t[n_, 0] := 2n + 1; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import accumulate, count, islice
def A000754_gen(): # generator of terms
    blist = tuple()
    for i in count(1,2):
        yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
A000754_list = list(islice(A000754_gen(),40)) # Chai Wah Wu, Jun 12 2022

From Reinhard Zumkeller, Nov 02 2013: (Start)
a(n) = Sum_{k=0..n} A116666(n+1,k)*A000111(n-k).
a(n) = Sum_{k=0..n} A109449(n,k)*(2*k + 1). (End)
E.g.f.: (sec(x) + tan(x))*exp(x)*(2*x + 1). - Sergei N. Gladkovskii, Oct 30 2014
a(n) ~ n! * (Pi+1) * exp(Pi/2) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Oct 30 2014

A029885 Boustrophedon transform of 1 followed by Thue-Morse sequence A001285.

Original entry on oeis.org

1, 2, 5, 13, 34, 108, 415, 1841, 9381, 53733, 342086, 2395481, 18300250, 151453434, 1349856656, 12890177378, 131298281746, 1420980348324, 16283235530691, 196958363484995, 2507751773736087, 33526171616091612

Offset: 0

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform.
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A109449, A001285, A230958, A230950, A230951.

a029885 n = sum $ zipWith (*) (a109449_row n) (1 : map fromIntegral a001285_list)
-- Reinhard Zumkeller, Nov 04 2013

tm[n_] := Mod[Sum[Mod[Binomial[n, k], 2], {k, 0, n}], 3];
T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
a[n_] := Sum[T[n, k] If[k == 0, 1, tm[k - 1]], {k, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 02 2019 *)

Definition corrected by Reinhard Zumkeller, Nov 04 2013

A230950 Boustrophedon transform of Thue-Morse sequence A010060.

Original entry on oeis.org

0, 1, 3, 6, 15, 50, 186, 834, 4243, 24318, 154780, 1083952, 8280624, 68531308, 610796150, 5832677415, 59411150931, 642979374958, 7368000716808, 89121684577460, 1134732527849730, 15170256449030866, 212469074496520610, 3111026318662704255, 47532980801984327584

Offset: 0

Reinhard Zumkeller, Nov 03 2013

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
Wikipedia, Boustrophedon transform
Index entries for sequences related to boustrophedon transform

Cf. A029885, A230951, A230952.

Cf. A010060, A109449.

a230950 n = sum $ zipWith (*) (a109449_row n) $ map fromIntegral a010060_list

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
a[n_] := Sum[T[n, k] ThueMorse[k], {k, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jul 02 2019 *)

from itertools import count, islice, accumulate
def A230950_gen(): # generator of terms
    blist = tuple()
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist), initial=i.bit_count()&1)))[-1]
A230950_list = list(islice(A230950_gen(),30)) # Chai Wah Wu, Apr 17 2023

a(n) = Sum_{k=0..n} A109449(n,k)*A010060(k).

A230951 Boustrophedon transform of Thue-Morse sequence A010059.

Original entry on oeis.org

1, 1, 1, 3, 9, 27, 108, 475, 2421, 13859, 88254, 617957, 4720980, 39070669, 348225424, 3325303894, 33871280413, 366573108019, 4200618576106, 50809739256049, 646929695900154, 8648812936664311, 121132117170628444, 1773647319453218254, 27099334868109293640

Offset: 0

Reinhard Zumkeller, Nov 03 2013

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
Wikipedia, Boustrophedon transform
Index entries for sequences related to boustrophedon transform

Cf. A029885, A230950, A230952.

Cf. A010059, A109449.

a230951 n = sum $ zipWith (*) (a109449_row n) $ map fromIntegral a010059_list

from itertools import count, islice, accumulate
def A230951_gen(): # generator of terms
    blist = tuple()
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist), initial=i.bit_count()&1^1)))[-1]
A230951_list = list(islice(A230951_gen(),30)) # Chai Wah Wu, Apr 17 2023

a(n) = Sum_{k=0..n} A109449(n,k)*A010059(k).

A000697 Boustrophedon transform of 1, 1, 4, 9, 16, ...

Original entry on oeis.org

1, 2, 7, 26, 89, 316, 1243, 5564, 28321, 162160, 1032051, 7226636, 55206161, 456886912, 4072080587, 38885496092, 396084390849, 4286637591872, 49121248360291, 594159600856332, 7565074996215025, 101137602761945440

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform.
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996), 44-54 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A000290, A000745, A109449.

a000697 n = sum $ zipWith (*) (a109449_row n) (1 : tail a000290_list)
-- Reinhard Zumkeller, Nov 04 2013

t[n_, 0] := If[n==0, 1, n^2]; t[n_, k_] := t[n, k] = t[n, k-1]+t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import accumulate, count, islice
def A000697_gen(): # generator of terms
    yield 1
    blist, m = (1,), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m += 2*i+1
A000697_list = list(islice(A000697_gen(),40)) # Chai Wah Wu, Jun 12 2022

E.g.f.: (1 + exp(x)*x*(1 + x))*(sec(x) + tan(x)). - Sergei N. Gladkovskii, Oct 29 2014

a(n) ~ n! * (4 + exp(Pi/2)*Pi*(2 + Pi)) * 2^n / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

cp's OEIS Frontend

A247453 T(n,k) = binomial(n,k)*A000111(n-k)*(-1)^(n-k), 0 <= k <= n.

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A000718 Boustrophedon transform of triangular numbers 1,1,3,6,10,...

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A000732 Boustrophedon transform of 1 & primes: 1,2,3,5,7,...

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A000734 Boustrophedon transform of 1,1,2,4,8,16,32,...

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Sage

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A000751 Boustrophedon transform of partition numbers.

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A000754 Boustrophedon transform of odd numbers.

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Haskell

Mathematica

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A029885 Boustrophedon transform of 1 followed by Thue-Morse sequence A001285.

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A247453 T(n,k) = binomial(n,k)A000111(n-k)(-1)^(n-k), 0 <= k <= n.