A100100 -id:A100100 - cp's OEIS frontend

A110576 Sequence is {a(0,n)}, where a(m,0)=1, a(m,n) = a(m,n-1) + a(m-1,n), a(0,n+1) = a(n,n).

1, 1, 2, 7, 29, 132, 648, 3407, 19109, 113946, 719896, 4802318, 33712717, 248285282, 1912928549, 15379305080, 128729241725, 1119519156562, 10097102345993, 94285391374568, 910145431036423, 9069616636456648, 93179779321299479

Offset: 0

Leroy Quet, Jul 28 2005

Equals eigensequence of triangle A100100. - Gary W. Adamson, Feb 02 2009

			a(0,n): 1, 1, 2, 7, 29
a(1,n): 1, 2, 4, 11
a(2,n): 1, 3, 7, 18
a(3,n): 1, 4, 11, 29
Since a(3,3) = 29, a(0,4) also is 29.

G. C. Greubel, Table of n, a(n) for n = 0..570

Cf. A110577, A110578, A110579, A110580.

Cf. A100100. - Gary W. Adamson, Feb 02 2009

a[0, 0] := 1; a[0, 1] := 1; a[0, n_] := a[0, n] = Sum[Binomial[2*n - k - 3, n - 2]*a[0, k], {k, 0, n - 1}]; Table[a[0,n], {n,0,50}] (* G. C. Greubel, Aug 31 2017 *)

a(0, n+1) = Sum_{k=0..n} binomial(2*n-k-1, n-1)*a(0, k), with a(0,0) = 1.

More terms from Ryan Propper, Sep 25 2005

A262737 O.g.f. exp( Sum_{n >= 1} A262732(n)*x^n/n ).

Original entry on oeis.org

1, 8, 95, 1336, 20642, 338640, 5791291, 102108760, 1842857390, 33879118384, 632210693270, 11944142806064, 228010741228740, 4391334026631072, 85221618348230355, 1664901954576830360, 32716286416687895862, 646228961799752926320, 12823701194384778672322

Offset: 0

Peter Bala, Sep 29 2015

O.g.f. is 1/x * the series reversion of x*(1 - x)^k/(1 + x)^(k+2) for k = 3. See the cross references for related sequences obtained from other values of k.

Peter Bala, Notes on logarithmic differentiation, the binomial transform and series reversion

Cf. A000108 (k = 0), A007297 (k = 1), A066357 (k = 2), A262738 (k = 4), A262739 (k = 5), A262740 (k = 6), A262732.

A262737 := proc (n) option remember; if n = 0 then 1 else add(1/k!*(5*k)!/GAMMA(5*k/2 + 1)*GAMMA(3*k/2 + 1)/(3*k)!*A262737(n-k), k = 1 .. n)/n end if; end proc:
seq(A262737(n), n = 0 .. 20);

a(n) = sum(k=0, n, binomial(5*(n+1),k)*binomial(4*(n+1)-k-2,(n+1)-k-1))/(n+1); \\ Altug Alkan, Oct 03 2015

a(n-1) = 1/n * Sum_{i = 0..n-1} binomial(5*n,i)*binomial(4*n-i-2,n-i-1).
O.g.f.: A(x) = exp ( Sum_{n >= 1} 1/n! * (5*n)!/(5*n/2)! * (3*n/2)!/(3*n)!*x^n/n ) = 1 + 8*x + 195*x^2 + 1336*x^3 + ....
1 + x*A'(x)/A(x) is the o.g.f. for A262732.
O.g.f. is the series reversion of x*(1 - x)^3/(1 + x)^5.
a(0) = 1 and for n >= 1, a(n) = 1/n * Sum {k = 1..n} 1/k!*(5*k)!/GAMMA(5*k/2+1)*GAMMA(3*k/2+1)/(3*k)! * a(n-k).

A262738 O.g.f. exp( Sum_{n >= 1} A211419(n)*x^n/n ).

Original entry on oeis.org

1, 10, 149, 2630, 51002, 1050132, 22539085, 498732014, 11296141454, 260613866380, 6103074997890, 144696786555580, 3466352150674324, 83776927644646952, 2040261954214847421, 50018542073019175806, 1233419779839067305350, 30572886836581693309020

Offset: 0

Peter Bala, Sep 29 2015

O.g.f. is 1/x * the series reversion of x*(1 - x)^k/(1 + x)^(k+2) at k = 4. See the cross references for related sequences obtained from other values of k.

Peter Bala, Notes on logarithmic differentiation, the binomial transform and series reversion

Cf. A211419, A000108 (k = 0), A007297 (k = 1), A066357 (k = 2), A262737 (k = 3), A262739 (k = 5), A262740 (k = 6).

A262738 := proc(n) option remember; if n = 0 then 1 else add((6*k)!*(2*k)!/((4*k)!*(3*k)!*k!)*A262738(n-k), k = 1 .. n)/n end if; end proc:
seq(A262738(n), n = 0..20);

a(n) = sum(k=0, n, binomial(6*(n+1),k)*binomial(5*(n+1)-k-2,(n+1)-k-1))/(n+1); \\ Altug Alkan, Oct 03 2015

a(n-1) = 1/n * Sum_{i = 0..n-1} binomial(6*n,i)*binomial(5*n-i-2,n-i-1).
O.g.f.: A(x) = exp ( Sum_{n >= 1} (6*n)!*(2*n)!/((4*n)!*(3*n)!*n!)*x^n/n ) = 1 + 10*x + 149*x^2 + 2630*x^3 + ....
1 + x*A'(x)/A(x) is the o.g.f. for A211419.
O.g.f. is the series reversion of x*(1 - x)^4/(1 + x)^6.
a(0) = 1 and for n >= 1, a(n) = 1/n * Sum {k = 1..n} (6*k)!*(2*k)!/((4*k)!*(3*k)!*k!)*a(n-k).

A262739 O.g.f. exp( Sum_{n >= 1} A262733(n)*x^n/n ).

Original entry on oeis.org

1, 12, 215, 4564, 106442, 2635704, 68031147, 1810302340, 49308457334, 1368019979976, 38525145673126, 1098380420669000, 31641932951483220, 919622628946689648, 26931762975278938035, 793967020231145502564, 23543663463050594677310, 701763102761640853890600, 21014048069544552257072530, 631868353403527700756671320, 19070677448561228207945931276

Offset: 0

Peter Bala, Sep 29 2015

O.g.f. is 1/x * the series reversion of x*(1 - x)^k/(1 + x)^(k+2) when k = 5. See the cross references for related sequences obtained from other values of k.

Peter Bala, Notes on logarithmic differentiation, the binomial transform and series reversion

Cf. A262733, A211419, A000108 (k = 0), A007297 (k = 1), A066357 (k = 2), A262737 (k = 3), A262738 (k = 4), A262740 (k = 6).

A262739 := proc (n) option remember; if n = 0 then 1 else add(1/k!*(7*k)!/GAMMA(7*k/2 + 1)*GAMMA(5*k/2 + 1)/(5*k)!*A262739(n-k), k = 1 .. n)/n end if; end proc:
seq(A262739(n), n = 0..20);

a(n) = sum(k=0, n, binomial(7*(n+1),k)*binomial(6*(n+1)-k-2,(n+1)-k-1))/(n+1); \\ Altug Alkan, Oct 03 2015

a(n-1) = 1/n * Sum_{i = 0..n-1} binomial(7*n,i)*binomial(6*n-i-2,n-i-1).
O.g.f.: A(x) = exp ( Sum_{n >= 1} 1/n! * (7*n)!/(7*n/2)! * (5*n/2)!/(5*n)!*x^n/n ) = 1 + 12*x + 215*x^2 + 4564*x^3 + ....
1 + x*A'(x)/A(x) is the o.g.f. for A262733.
O.g.f. is the series reversion of x*(1 - x)^5/(1 + x)^7,
a(0) = 1 and for n >= 1, a(n) = 1/n * Sum {k = 1..n} 1/k! * (7*k)!/(7*k/2)! * (5*k/2)!/(5*k)!*a(n-k).

A262740 O.g.f. exp( Sum_{n >= 1} A211421(n)*x^n/n ).

Original entry on oeis.org

1, 14, 293, 7266, 197962, 5726364, 172662765, 5367187226, 170772853790, 5534640052292, 182070248073826, 6063785526898644, 204055962203476788, 6927718839334775608, 236994877398511998717, 8161492483543100398410, 282705062046649346154006, 9843330120848835962213940

Offset: 0

Peter Bala, Sep 29 2015

O.g.f. is 1/x * the series reversion of x*(1 - x)^k/(1 + x)^(k+2) at k = 6. See the cross references for related sequences obtained from other values of k.

Peter Bala, Notes on logarithmic differentiation, the binomial transform and series reversion

Cf. A211421, A000108 (k = 0), A007297 (k = 1), A066357 (k = 2), A262737 (k = 3), A262738 (k = 4), A262739 (k = 5).

#A262740
A262740 := proc (n) option remember; if n = 0 then 1 else add(1/k!*(8*k)!/(4*k)!*(3*k)!/(6*k)!*A262740(n-k), k = 1 .. n)/n end if; end proc:
seq(A262740(n), n = 0..17);

a(n) = sum(k=0, n, binomial(8*(n+1),k)*binomial(7*(n+1)-k-2,(n+1)-k-1))/(n+1); \\ Altug Alkan, Oct 03 2015

a(n-1) = 1/n * Sum_{i = 0..n-1} binomial(8*n,i)*binomial(7*n-i-2,n-i-1).
O.g.f.: A(x) = exp ( Sum_{n >= 1} 1/n! * (8*n)!/(4*n)! * (3*n)!/(6*n)!*x^n/n ) = 1 + 14*x + 293*x^2 + 7266*x^3 + ....
1 + x*A'(x)/A(x) is the o.g.f. for A211421.
O.g.f. is the series reversion of x*(1 - x)^6/(1 + x)^8.
a(0) = 1 and for n >= 1, a(n) = 1/n * Sum {k = 1..n} 1/k! * (8*k)!/(4*k)! * (3*k)!/(6*k)!*a(n-k).

A262145 O.g.f.: exp( Sum_{n >= 1} A000182(n+1)*x^n/n ), where A000182 is the sequence of tangent numbers.

Original entry on oeis.org

1, 2, 10, 108, 2214, 75708, 3895236, 280356120, 26824493574, 3287849716332, 501916845156012, 93337607623037544, 20766799390944491100, 5446109742113077482456, 1662395457873577922274888

Offset: 0

Peter Bala, Sep 13 2015

It appears that the sequence has integer entries. Calculation suggests the following conjecture: the expansion of exp( Sum_{n >= 1} A000182(n + m)*x^n/n ) has integer coefficients for m = 1, 2, 3, .... This is the case m = 1. Cf. A255881 and A255895.

First row of square array A262144.

P. Bala, Notes on logarithmic differentiation, the binomial transform and series reversion

Cf. A000182, A255881, A255895, A262144 (first row).

#A262145
#define tangent numbers A000182
A000182 := n -> (1/2) * 2^(2*n) * (2^(2*n) - 1) * abs(bernoulli(2*n))/n:
a := proc (n) option remember;
if n = 0 then 1 else
  add(A000182(k+1)*a(n-k), k = 1 .. n)/n
end if;
end proc:
seq(a(n), n = 0 .. 15);

max = 15; CoefficientList[E^Sum[(-1)^n*2^(2*n+1)*(4^(n+1)-1)*BernoulliB[2*(n+1)]*x^n / (n*(n+1)), {n, 1, max}] + O[x]^max, x] (* Jean-François Alcover, Sep 18 2015 *)

def a_list(n):
    T = [0]*(n+2); T[1] = 1
    for k in range(2, n+1): T[k] = (k-1)*T[k-1]
    for k in range(2, n+1):
        for j in range(k, n+1): T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]
    @cached_function
    def a(n): return sum(T[k+1]*a(n-k) for k in (1..n))//n if n> 0 else 1
    return [a(k) for k in range(n)]
a_list(15) # Peter Luschny, Sep 18 2015

Recurrence: a(n) = 1/n * Sum_{k = 1..n} A000182(k+1)*a(n-k).

A100217 Diagonal sums of a binomial number triangle.

Original entry on oeis.org

1, 1, 4, 12, 42, 149, 543, 2007, 7501, 28265, 107196, 408653, 1564506, 6010964, 23164467, 89501021, 346588092, 1344804060, 5227147969, 20349230347, 79330194097, 309653982738, 1210071825851, 4733665388134, 18535196846866

Offset: 0

Paul Barry, Nov 08 2004

Diagonal sums of A100100.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A100100.

[(&+[Binomial(2*n-3*k-1, n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Mar 28 2024

A100217[n_]:= Sum[Binomial[2*n-3*k-1,n-2*k], {k,0,Floor[n/2]}];
Table[A100217[n], {n,0,40}] (* G. C. Greubel, Mar 28 2024 *)

[sum(binomial(2*n-3*k-1, n-2*k) for k in range(1+n//2)) for n in range(41)] # G. C. Greubel, Mar 28 2024

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1, n-2*k).

A262144 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} d(n,i+1)*x^i/i ) for n >= 1, where d(n,k) is Shanks's array of generalized Euler and class numbers.

Original entry on oeis.org

1, 1, 2, 1, 11, 10, 1, 46, 241, 108, 1, 128, 2739, 10411, 2214, 1, 272, 16384, 265244, 836321, 75708, 1, 522, 64964, 2883584, 45094565, 112567243, 3895236, 1, 904, 212325, 18852096, 822083584, 12975204810, 22949214033

Offset: 1

Peter Bala, Sep 18 2015

Shanks's array d(n,k) n >= 1, k >= 1, is A235606.
We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 1, 2, ... and for each n >= 1, the expansion of exp( Sum_{i >= 1} d(n,i + r)*x^i/i ) has integer coefficients. This is the case r = 1.
For the similarly defined array associated with Shanks' c(n,k) array see A262143.

			The triangular array begins
1
1   2
1  11     10
1  46    241      108
1 128   2739    10411      2214
1 272  16384   265244    836321       75708
1 522  64964  2883584  45094565   112567243     3895236
1 904 212325 18852096 822083584 12975204810 22949214033 ...
The square array begins (row indexing n starts at 1)
1, 2, 10, 108, 2214, 75708, 3895236, 280356120, 26824493574, ...
1, 11, 241, 10411, 836321, 112567243, 22949214033, 6571897714923, 2507281057330113, ...
1, 46, 2739, 265244, 45094565, 12975204810, 5772785327575, 3656385436507960, 3107332328608143945, ...
1, 128, 16384, 2883584, 822083584, 395136991232, 300338473074688, 330739694704787456, 493338658405976375296, ...
1, 272, 64864, 18852096, 8133183744, 5766226378752, 6562478680375296, 11019751545852395520, 25333348417380699340800, ...
1, 522, 212325, 94501768, 57064909374, 54459242196516, 84430282319806062, 197625548666434041000, 642556291067409622713543, ...
1, 904, 586452, 382674008, 311514279098, 379982635729752, 753288329161251844, 2308779464340711480136, 10003494921382094286802995, ...

P. Bala, Notes on logarithmic differentiation, the binomial transform and series reversion
William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia , The generating function for the Dirichlet series Lm(s) Mathematics of Computation, Vol. 81, No. 278, April 2012.
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Cf. A000182 (d(1,n)), A000464 (d(2,n)), A000191 (d(3,n)), A000318 (d(4,n)), A000320 (d(5,n)), A000411 (d(6,n)), A064072 (d(7,n)), A235605, A235606, A262143, A262145 (row 1 of square array).

A307495 Expansion of Sum_{k>=0} k!((1 - sqrt(1 - 4x))/2)^k.

Original entry on oeis.org

1, 1, 3, 12, 57, 312, 1950, 13848, 111069, 998064, 9957186, 109305240, 1309637274, 17006109072, 237888664572, 3566114897520, 57030565449765, 969154436550240, 17439499379433690, 331268545604793240, 6624013560942038670, 139080391965533653200, 3059323407592802838180, 70355685298375014175440

Offset: 0

Ilya Gutkovskiy, Apr 10 2019

nonn

Catalan transform of A000142 (factorial numbers).
From Peter Bala, Jan 27 2020: (Start)
This sequence is the main diagonal of the lower triangular array formed by putting the sequence of factorial numbers in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.
    1
    1    1
    2    3    3
    6    9   12   12
   24   33   45   57   57
  120  153  198  255  312  312
  ...
Alternatively, the sequence can be obtained by multiplying the sequence of factorial numbers by the array A106566.
(End)

P. Bala, A note on the Catalan transform of a sequence

Cf. A000108, A000142, A013999, A100100, A106566, A307496.

nmax = 23; CoefficientList[Series[Sum[k! ((1 - Sqrt[1 - 4 x])/2)^k, {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1/(1 + ContinuedFractionK[-Floor[(k + 1)/2] (1 - Sqrt[1 - 4 x])/2, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
Join[{1}, Table[1/n Sum[Binomial[2n - k - 1, n - k] k k!, {k, n}], {n, 23}]]

G.f.: 1 /(1 - x*c(x)/(1 - x*c(x)/(1 - 2*x*c(x)/(1 - 2*x*c(x)/(1 - 3*x*c(x)/(1 - 3*x*c(x)/(1 - ...))))))), a continued fraction, where c(x) = g.f. of Catalan numbers (A000108).
Sum_{n>=0} a(n)*(x*(1 - x))^n = g.f. of A000142.
a(n) = (1/n) * Sum_{k=1..n} binomial(2*n-k-1,n-k)*k*k! for n > 0.
a(n) ~ exp(1) * n!. - Vaclav Kotesovec, Aug 10 2019

A191528 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n that have k returns to the axis.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 1, 6, 3, 1, 10, 6, 3, 1, 20, 10, 4, 1, 35, 20, 10, 4, 1, 70, 35, 15, 5, 1, 126, 70, 35, 15, 5, 1, 252, 126, 56, 21, 6, 1, 462, 252, 126, 56, 21, 6, 1, 924, 462, 210, 84, 28, 7, 1, 1716, 924, 462, 210, 84, 28, 7, 1, 3432, 1716, 792, 330, 120, 36, 8, 1, 6435, 3432, 1716, 792, 330, 120, 36, 8, 1

Offset: 0

Emeric Deutsch, Jun 06 2011

Row n contains 1+floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) =A001405(n).
T(n,0) = A001405(n-1).
Rows 0, 2, 4, ... form triangle A100100.
Rows 1, 3, 5, ... form triangle A092392.
Sum_{k>=0} k*T(n,k) = A037955(n).
From Roger Ford, Oct 16 2020: (Start)
This is an empirical observation. T(n,k) = the number of different semi-meander arch depth models with n+2 top arches and k+1 arches at depth 0.  T(3,1) = the number of different semi-meander arch depth models with 5 top arches and 2 arches at depth 0.
Example: The depth of a semi-meander arch is the number of covering arches directly above the arch.  The arch depth model is the number of arches at each depth starting at 0 for a specific semi-meander.  The following is the arch depth models for semi-meanders with 5 top arches.
            /\                                   /\
           //\\                                 /  \
          ///\\\             depth             //\  \             depth
         ////\\\\  /\       (0)(1)(2)(3)      ///\\/\\ /\        (0)(1)(2)
depth    0123      0  model= 2  1  1  1       012  1   0   model= 2  2  1
         /\
        //\\    /\         depth         /\   /\            depth
       ///\\\  //\\       (0)(1)(2)     //\\ //\\ /\       (0)(1)
depth  012     01   model= 2  2  1      01   01   0  model= 3  2
         /\
        /  \               depth
       //\/\\ /\ /\       (0)(1)
depth  01 1   0  0  model= 3  2
There are 5 more semi-meanders with 5 top arches.  They are reflections of the above semi-meanders over a center vertical line and they yield the same arch depth models as the semi-meanders above.
T(3,1) = 2    different models=  2 2 1 and 2 1 1 1;
T(3,2) = 1     model=  3 2    (End).

			T(6,2)=3 because we have U(D)U(D)UU, U(D)UUD(D), and UUD(D)U(D), where U=(1,1) and D=(1,-1) (the return steps to the axis are shown between parentheses).
Triangle starts:
   1:
   1;
   1, 1;
   2, 1;
   3, 2, 1;
   6, 3, 1;
  10, 6, 3, 1;

Indranil Ghosh, Rows 0..100, flattened

Cf. A001405, A037955, A092392, A100100.

T := proc (n, k) if k <= floor((1/2)*n) then binomial(n-k-1, ceil((1/2)*n)-1) else 0 end if end proc: for n from 0 to 16 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Flatten[Table[Binomial[n-k-1,Ceiling[(n/2)-1]],{n,0,16},{k,0,Floor[n/2]}]] (* Indranil Ghosh, Mar 05 2017 *)

tabf(nn) = if(n==0, print1(1,", "), {for (n=1, nn, for(k=0, floor(n/2), print1(binomial(n-k-1, ceil((n/2)-1)),", ");); print();); });
tabf(16); \\ Indranil Ghosh, Mar 05 2017

T(n,k) = binomial(n-k-1, ceiling(n/2)-1) if 0 <= k <= floor(n/2).

G.f.: G(t,z) = 1/((1-z*c)*(1-t*z^2*c)), where c = (1-sqrt(1-4*z^2))/(2*z^2) is the Catalan function with argument z^2.

cp's OEIS Frontend

A110576 Sequence is {a(0,n)}, where a(m,0)=1, a(m,n) = a(m,n-1) + a(m-1,n), a(0,n+1) = a(n,n).

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A262737 O.g.f. exp( Sum_{n >= 1} A262732(n)*x^n/n ).

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A262738 O.g.f. exp( Sum_{n >= 1} A211419(n)*x^n/n ).

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A262739 O.g.f. exp( Sum_{n >= 1} A262733(n)*x^n/n ).

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A262740 O.g.f. exp( Sum_{n >= 1} A211421(n)*x^n/n ).

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A262145 O.g.f.: exp( Sum_{n >= 1} A000182(n+1)*x^n/n ), where A000182 is the sequence of tangent numbers.

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A100217 Diagonal sums of a binomial number triangle.

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Mathematica

SageMath

A307495 Expansion of Sum_{k>=0} k!((1 - sqrt(1 - 4x))/2)^k.