A099237 -id:A099237 - cp's OEIS frontend

A099233 Square array read by antidiagonals associated to sections of 1/(1-x-x^k).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 8, 1, 1, 1, 6, 15, 26, 28, 13, 1, 1, 1, 7, 21, 45, 69, 60, 21, 1, 1, 1, 8, 28, 71, 140, 181, 129, 34, 1, 1, 1, 9, 36, 105, 251, 431, 476, 277, 55, 1, 1, 1, 10, 45, 148, 413, 882, 1326, 1252, 595, 89, 1

Offset: 0

Paul Barry, Oct 08 2004

			Rows begin
  1, 1, 1,  1,  1,   1, ...
  1, 1, 2,  3,  5,   8, ...
  1, 1, 3,  6, 13,  28, ...
  1, 1, 4, 10, 26,  69, ...
  1, 1, 5, 15, 45, 140, ...
Row 1 is the 0-section of 1/(1-x-x)   (A000079);
Row 2 is the 1-section of 1/(1-x-x^2) (A000045);
Row 3 is the 2-section of 1/(1-x-x^3) (A000930);
Row 4 is the 3-section of 1/(1-x-x^4) (A003269);
etc.

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Sums of antidiagonals are A099236.
Columns include A000217, A008778.
Rows include A000045, A002478, A099234, A099235.
Main diagonal gives A099237.
Cf. A099238.

Square array T(n, k) = Sum_{j=0..n} binomial(k(n-j), j).

Rows are generated by 1/(1-x(1+x)^k) and satisfy a(n) = Sum_{k=0..n} binomial(n, k)a(n-k-1).

A361281 a(n) = n! * Sum_{k=0..n} binomial(n*k,n-k)/k!.

Original entry on oeis.org

1, 1, 5, 37, 481, 10001, 288901, 10820965, 511186817, 29843419681, 2106779832901, 176180844038981, 17165338119936865, 1924030148121500017, 245630480526435293381, 35409038825312233143301, 5719025066628373334423041, 1027649751647068260334391105

Offset: 0

Seiichi Manyama, Mar 06 2023

nonn

From  Peter Bala, Mar 12 2023: (Start)
It appears that a(n) == 1 (mod 4) and a(5*n+2) == 0 (mod 5) for all n. More generally we conjecture that a(n+k) == a(n) (mod k) for all n and k. If true, then for each k, the sequence a(n) taken modulo k is a periodic sequence and the period divides k.
Let F(x) and G(x) be power series with integer coefficients with G(0) = 1. Define b(n) = n! * [x^n] F(x)*exp(x*G(x)^n). Then we conjecture that b(n+k) == b(n) (mod k) for all n and k. The present sequence is the case F(x) = 1, G(x) = 1 + x. Cf. A278070. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..275

Main diagonal of A361277.

Cf. A096131, A099237, A226391, A278070, A293013.

a(n) = n!*sum(k=0, n, binomial(n*k, n-k)/k!);

a(n) = n! * [x^n] exp(x * (1+x)^n).

log(a(n)) ~ n*(2*log(n) - log(log(n)) - 1 - log(2) + log(log(n))/log(n) + 1/(2*log(n)) + log(2)/log(n) - 1/(8*log(n)^2)). - Vaclav Kotesovec, Mar 12 2023

A362079 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^n)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 10, 0, 1, 4, 12, 28, 45, 0, 1, 5, 18, 55, 145, 251, 0, 1, 6, 25, 92, 315, 896, 1624, 0, 1, 7, 33, 140, 571, 2106, 6328, 11908, 0, 1, 8, 42, 200, 930, 4076, 15946, 50212, 97545, 0, 1, 9, 52, 273, 1410, 7026, 32718, 134730, 441489, 880660, 0

Offset: 0

Seiichi Manyama, Apr 08 2023

			Square array begins:
  1,   1,   1,    1,    1,    1, ...
  0,   1,   2,    3,    4,    5, ...
  0,   3,   7,   12,   18,   25, ...
  0,  10,  28,   55,   92,  140, ...
  0,  45, 145,  315,  571,  930, ...
  0, 251, 896, 2106, 4076, 7026, ...

Columns k=0..3 give A000007, A099237, A362084, A362085.
Main diagonal gives A362080.
Cf. A362078.

T(n, k) = sum(j=0, n, binomial(j+k-1, j)*binomial(n*j, n-j));

T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-k,j) * binomial(n*j,n-j) = Sum_{j=0..n} binomial(j+k-1,j) * binomial(n*j,n-j).

A157114 Triangle T(n, k) = binomial(nk, n-k) + binomial(n(n-k), k), read by rows.

Original entry on oeis.org

2, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 56, 16, 1, 1, 25, 225, 225, 25, 1, 1, 36, 771, 1632, 771, 36, 1, 1, 49, 2597, 9261, 9261, 2597, 49, 1, 1, 64, 9136, 52384, 71920, 52384, 9136, 64, 1, 1, 81, 33777, 320814, 525987, 525987, 320814, 33777, 81, 1, 1, 100, 129130, 2090540, 4326015, 4237520, 4326015, 2090540, 129130, 100, 1

Offset: 0

Roger L. Bagula, Feb 23 2009

			Triangle begins as:
  2;
  1,   1;
  1,   4,      1;
  1,   9,      9,       1;
  1,  16,     56,      16,       1;
  1,  25,    225,     225,      25,       1;
  1,  36,    771,    1632,     771,      36,       1;
  1,  49,   2597,    9261,    9261,    2597,      49,       1;
  1,  64,   9136,   52384,   71920,   52384,    9136,      64,      1;
  1,  81,  33777,  320814,  525987,  525987,  320814,   33777,     81,   1;
  1, 100, 129130, 2090540, 4326015, 4237520, 4326015, 2090540, 129130, 100, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A099237.

A157114:= func< n,k | Binomial(n*k, n-k) + Binomial(n*(n-k), k) >;
[A157114(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

A157114:= (n,k) -> binomial(n*k, n-k) + binomial(n*(n-k), k);
seq(seq(A157114(n,k), k=0..n), n=0..12); # G. C. Greubel, Mar 09 2021

T[n_, k_]:= Binomial[n*k, n-k], Binomial[n*(n-k), k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Mar 09 2021 *)

def A157114(n,k): return binomial(n*k, n-k) + binomial(n*(n-k), k)
flatten([[A157114(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021

T(n, k) = binomial(n*k, n-k) + binomial(n*(n-k), k).

Sum_{k=0..n} T(n,k) = 2*A099237(n). - G. C. Greubel, Mar 09 2021

Edited by G. C. Greubel, Mar 09 2021

A299044 G.f. Sum_{n>=0} Series_Reversion( x/(1+x)^n )^n.

Original entry on oeis.org

1, 1, 2, 6, 25, 129, 784, 5472, 42993, 374190, 3564176, 36808647, 409067204, 4861490200, 61457674398, 822732344816, 11618029697489, 172476856415121, 2683881876383377, 43660291710726058, 740764460615030663, 13080604188895285878, 239939914279952537597, 4564083798329838120034, 89886989241387131773525, 1830230258908641519168564

Offset: 0

Paul D. Hanna, Feb 18 2018

nonn

Antidiagonal sums of square table A299427.

			G.f. A(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 129*x^5 + 784*x^6 + 5472*x^7 + 42993*x^8 + 374190*x^9 + 3564176*x^10 + ...
such that
A(x) = 1 + x*R(x,1) + x^2*R(x,2)^4 + x^3*R(x,3)^9 + x^4*R(x,4)^16 + x^5*R(x,5)^25 + x^6*R(x,6)^36 + ...
where series R(x,n) = 1 + x*R(x,n)^n begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + ...
R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ...
R(x,3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...
R(x,4) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ...
R(x,5) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ...
R(x,6) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + ...
...
and series R(x,n)^(n^2) begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...
R(x,2)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 165*x^4 + 572*x^5  + ...
R(x,3)^9 = 1 + 9*x + 63*x^2 + 408*x^3 + 2565*x^4 + 15939*x^5 + ...
R(x,4)^16 = 1 + 16*x + 184*x^2 + 1872*x^3 + 17980*x^4 + 167552*x^5 + ...
R(x,5)^25 = 1 + 25*x + 425*x^2 + 6175*x^3 + 82775*x^4 + 1059380*x^5 + ...
R(x,6)^36 = 1 + 36*x + 846*x^2 + 16536*x^3 + 292581*x^4 + 4874688*x^5 + ...
...
demonstrating that A(x) = Sum_{n>=1} x^n * R(x,n)^(n^2).

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A299427, A099237, A299043.

a[0]=1; a[n_] := Sum[Binomial[n*(n-k), k]*(n-k)/n, {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 23 2018 *)

{a(n) = my(A,Ox=x^2*O(x^n)); A = sum(m=0,n+1, serreverse( x/(1+x +Ox)^m +Ox)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{a(n) = if(n==0,1, sum(k=0,n, binomial(n*(n-k),k) * (n-k)/n ) )}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n equals the following expressions.
(1) A(x) = Sum_{n>=0} Series_Reversion( x/(1+x)^n )^n.
(2) A(x) = Sum_{n>=1} x^n * R(x,n)^(n^2), where
(2.a) R(x,n) = 1 + x*R(x,n)^n,
(2.b) R(x,n)^n = Series_Reversion( x/(1+x)^n ) / x,
(2.c) R(x,n)^n = Sum_{k>=0} C(n*(k+1), k)/(k+1) * x^k;
(2.d) R(x,n)^(n^2) = Sum_{k>=0} C(n*(n+k), k) * n/(n+k) * x^k.
FORMULAS FOR TERMS.
a(n) = Sum_{k=0..n} binomial(n*(n-k), k) * (n-k)/n.
a(n)^(1/n) ~ n^(n/w) * (n+1-w)^(1 - (n+1)/w) * (w-1)^(1/w - 1), where w = LambertW(exp(1)*n). - Vaclav Kotesovec, Feb 19 2018

A362080 a(n) = [x^n] 1/(1 - x*(1+x)^n)^n.

Original entry on oeis.org

1, 1, 7, 55, 571, 7026, 98925, 1562219, 27214867, 516646333, 10589130223, 232574622116, 5440521381816, 134859301929873, 3527034780915985, 96965997588549555, 2793286163779275779, 84076751617833902070, 2637677096916448507104

Offset: 0

Seiichi Manyama, Apr 08 2023

Seiichi Manyama, Table of n, a(n) for n = 0..449

Main diagonal of A362078.
Main diagonal of A362079.
Cf. A099237.

a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(n*k, n-k));

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

A362084 a(n) = Sum_{k=0..n} (-1)^k * binomial(-2,k) * binomial(n*k,n-k).

Original entry on oeis.org

1, 2, 7, 28, 145, 896, 6328, 50212, 441489, 4248370, 44306306, 496991848, 5959111223, 75977511442, 1025441134747, 14594189335496, 218290221112929, 3421314388169988, 56043004143343843, 957209642080023286, 17011439135301438016, 313980693855333453740

Offset: 0

Seiichi Manyama, Apr 08 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..493

Column k=2 of A362079.

Cf. A099237.

a(n) = sum(k=0, n, (k+1)*binomial(n*k, n-k));

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

a(n) = [x^n] 1/(1 - x*(1+x)^n)^2.

A361829 a(n) = Sum_{k=0..n} binomial(2k,k) binomial(n*k,n-k).

Original entry on oeis.org

1, 2, 10, 62, 486, 4482, 47106, 553226, 7152438, 100644194, 1527758136, 24839853326, 430045385424, 7888706328934, 152685931935634, 3106864307092950, 66253232332628166, 1476558925897693698, 34307420366092350048, 829217371825336147142

Offset: 0

Seiichi Manyama, Mar 26 2023

nonn

Main diagonal of A361830.

Cf. A099237, A361835.

Table[Sum[Binomial[2*k,k]*Binomial[n*k,n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 26 2023 *)

a(n) = sum(k=0, n, binomial(2*k, k)*binomial(n*k, n-k));

a(n) = [x^n] 1/sqrt(1 - 4*x*(1+x)^n).

log(a(n)) ~ n*(log(n) + (2*log(2) - 1)/log(n) - (1 - 1/log(n))*log(log(n) - 1)). - Vaclav Kotesovec, Mar 26 2023

A359842 a(n) = Sum_{k=0..n} binomial(n*k,n+k).

Original entry on oeis.org

1, 0, 1, 90, 13690, 3443275, 1308315371, 701623884514, 505274768721332, 470638793249281593, 550707386335951810915, 790898932162231992184327, 1367864138835420575101044139, 2804370191530797723173615407860, 6725366044028696102055907486691290

Offset: 0

Vaclav Kotesovec, Jan 15 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..205

Cf. A096131, A099237, A226391.

a := proc (n) option remember; add(binomial(n*k, n+k), k = 0..n) end:
seq(a(n), n = 0..20); # Peter Bala, Jan 16 2023

Table[Sum[Binomial[n*k, n+k], {k, 0, n}], {n, 0, 20}]

a(n) ~ binomial(n^2,2*n).
a(n) ~ exp(2*n-2) * n^(2*n - 1/2) / (sqrt(Pi) * 2^(2*n+1)).
From Peter Bala, Jan 19 2023: (Start)
Conjectures: a(2^k) == 0 (mod 2^(k-1)) and a(3^k) == 0 (mod 3^(k+2)) for k >= 2; a(p^k) == 0 (mod p^(k+1)) for all primes p >= 5.
Let m be a positive integer. Similar recurrences may hold for the sequence whose n-th term is given by Sum_{k = 0..n} binomial(m*n*k, n+k). Cf. A099237. (End)

A361836 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n*k,n-k).

Original entry on oeis.org

1, 1, -1, -2, 13, -29, -80, 1268, -7351, 13276, 245746, -3632793, 27451743, -63909390, -1752952501, 34899085656, -370619158447, 1779155624299, 23668687715473, -780307293795152, 12058261763444876, -107734052276914986, -180664717708949253, 30298196609011736398

Offset: 0

Seiichi Manyama, Mar 26 2023

Cf. A099237, A361835.

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n*k, n-k));

a(n) = [x^n] 1/(1 - x*(1-x)^n).

cp's OEIS Frontend

A099233 Square array read by antidiagonals associated to sections of 1/(1-x-x^k).

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A361281 a(n) = n! * Sum_{k=0..n} binomial(n*k,n-k)/k!.

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A362079 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^n)^k.

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A157114 Triangle T(n, k) = binomial(n*k, n-k) + binomial(n*(n-k), k), read by rows.

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A299044 G.f. Sum_{n>=0} Series_Reversion( x/(1+x)^n )^n.

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A362080 a(n) = [x^n] 1/(1 - x*(1+x)^n)^n.

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A362084 a(n) = Sum_{k=0..n} (-1)^k * binomial(-2,k) * binomial(n*k,n-k).

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A361829 a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(n*k,n-k).

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A157114 Triangle T(n, k) = binomial(nk, n-k) + binomial(n(n-k), k), read by rows.

A361829 a(n) = Sum_{k=0..n} binomial(2k,k) binomial(n*k,n-k).