A002064 -id:A002064 - cp's OEIS frontend

A131700 Sum of all n-digit Cullen numbers.

13, 90, 1443, 6658, 81923, 827395, 17956868, 157286403, 1434451971, 12884901891, 114353504259, 1005022347267, 8761733283843, 166026255794180, 1337006139375619, 11434920928870403, 97390341941886979, 1799188051134513156, 14231374822490767363, 119903836479112085507

Offset: 1

Parthasarathy Nambi, Sep 15 2007

			Sum of all 1-digit Cullen numbers is 1 + 3 + 9 = 13.
Sum of all 2-digit Cullen numbers is 25 + 65 = 90.
Sum of all 3-digit Cullen numbers is 161 + 385 + 897 = 1443.

E.W. Weisstein, Cullen Number

Cf. A002064.

digNum[n_] := Length @ IntegerDigits[n]; cullen[n_] := n * 2^n + 1; digCount = 0; sum = 0; cumsum = {}; Do[c = cullen[n]; If[digNum[c] > digCount, digCount++; AppendTo[cumsum, sum]]; sum += c, {n, 0, 65}]; Differences[cumsum] (* Amiram Eldar, Nov 30 2019 *)

More terms from Amiram Eldar, Nov 30 2019

A264751 Triangle read by rows: T(n,k) is the number of sequences of k <= n throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the k-th throw.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 9, 11, 4, 1, 14, 26, 19, 5, 1, 20, 50, 55, 29, 6, 1, 27, 85, 125, 99, 41, 7, 1, 35, 133, 245, 259, 161, 55, 8, 1, 44, 196, 434, 574, 476, 244, 71, 9, 1, 54, 276, 714, 1134, 1176, 804, 351, 89, 10, 1, 65, 375, 1110, 2058, 2562, 2190, 1275, 485, 109, 11

Offset: 1

Louis Rogliano, Nov 26 2015

By empirical observation: Sum of rows is A002064.

			Triangle begins:
  1
  1    2
  1    5    3
  1    9   11    4
  1   14   26   19    5
  1   20   50   55   29    6
  1   27   85  125   99   41    7
  1   35  133  245  259  161   55    8
  1   44  196  434  574  476  244   71    9
  1   54  276  714 1134 1176  804  351   89   10
  1   65  375 1110 2058 2562 2190 1275  485  109   11

Cyann Donnot, Antoine Genitrini, Yassine Herida, Unranking Combinations Lexicographically: an efficient new strategy compared with others, hal-02462764 [cs] / [cs.DS] / [math] / [math.CO], 2020.
Antoine Genitrini and Martin Pépin, Lexicographic unranking of combinations revisited, hal-03040740v2 [cs.DM] [cs.DS] [math.CO], 2020.

Columns are: A000012 (k=1), A000096 (k=2), A051925 (k=3), A215862 (k=4), A264750 (k=5).
Cf. A007318 (binomial(n-1,k-1) = number of sequences of k throws of an n-sided die in which the sum of the throws equals n).
See also A002064.

T[n_, k_] := Module[
{i, total = 0, part, perm},
part = IntegerPartitions[n, {k}];
perm = Flatten[Table[Permutations[part[[i]]], {i, 1, Length[part]}],      1];
For[i = 1, i <= Length[perm], i++,    total += n + 1 - perm[[i, k]]    ];
Return[total];   ]
(* The rows are obtained by: *)
g[n_] := Table[T[n,k], {k,1,n}]
(* And the triangle is obtained by: *)
Table[g[n],{n,1,number_of_rows_wanted}]

Sum_{k = 1..n} T(n,k)*k/n^k = ((n+1)/n)^(n-1) = expected value of k.
Lim_{n->infinity} (expected value of k) = e = 2.71828182845... - Jon E. Schoenfield, Nov 26 2015
T(n,k) = Sum_{i=k..n} i*binomial(i-2,k-2). - Danny Rorabaugh, Mar 04 2016
T(n,n-1) = 2*T(n-1,n-1) + T(n-1,n-2).
By empirical observation, g.f. for column k: (x-k)/(x-1)^(k+1).

A354272 Irregular triangle read by rows: coefficients of polynomials which are the product of all possible monic Littlewood polynomials of degree n.

Original entry on oeis.org

1, -1, 0, 1, 1, 0, -2, 0, -1, 0, -2, 0, 1, 1, 0, -4, 0, 2, 0, -4, 0, 15, 0, 8, 0, -36, 0, 8, 0, 15, 0, -4, 0, 2, 0, -4, 0, 1, 1, 0, -8, 0, 20, 0, -24, 0, 58, 0, -80, 0, -92, 0, 120, 0, 147, 0, 384, 0, -2108, 0, 880, 0, 3940, 0, -3096, 0, 2288, 0, -2136, 0, -1803, 0, -2136, 0, 2288, 0, -3096, 0, 3940, 0, 880, 0, -2108, 0, 384, 0, 147, 0, 120, 0, -92, 0, -80, 0, 58, 0, -24, 0, 20, 0, -8, 0, 1

Offset: 0

Gleb Ivanov, May 22 2022

			The triangle T(n, k) begins
n\k  1 2  3 4  5 6  7 8 9 10 11 12  13 14 15 16 17 18 19 20 21 22 23 24 25
0:   1
1:  -1 0  1
2:   1 0 -2 0 -1 0 -2 0 1
3:   1 0 -4 0  2 0 -4 0 15 0  8  0 -36  0  8  0 15  0 -4  0  2  0 -4  0  1
...
E.g., row 2: {1,0,-2,0,-1,0,-2,0,1} corresponds to polynomial 1-2x^2-x^4-2x^6+x^8.
Number of terms in each row equals A002064(n).

Wikipedia, Littlewood polynomial

Cf. A020985, A002064 (row lengths).

row(n) = { Vecrev(Vec(prod (k=2^n, 2^(n+1)-1, Pol(apply(d -> if (d, 1, -1), binary(k)))))) } \\ Rémy Sigrist, Jul 21 2022

from itertools import product
def mult_pol(s1, s2):
    res = [0]*(len(s1)+len(s2)-1)
    for o1,i1 in enumerate(s1):
        for o2,i2 in enumerate(s2):
            res[o1+o2] += i1*i2
    return res
out = []
for d in range(0, 5):
    startp = [1,]
    for i in product((1,-1),repeat = d):
        startp = mult_pol(startp, list(i)+[1,])
    out.extend(startp)
print(out)

A373867 Perfect powers of the form x^y + y^x, where x > 1 and y > 1.

Original entry on oeis.org

8, 32, 100, 512, 33554432, 36893488147419103232, 2923003274661805836407369665432566039311865085952, 78804012392788958424558080200287227610159478540930893335896586808491443542994421222828532509769831281613255980613632

Offset: 1

Gonzalo Martínez, Jun 21 2024

nonn

Subsequence of A076980: a(n) is a Leyland number that is a perfect power. The condition that x > 1 and y > 1 is necessary, otherwise every perfect power would belong to this sequence, since m^n = (m^n-1)^1 + 1^(m^n-1).
If x = y = 2^k, then x^y + y^x = 2^(k*2^k + 1) belongs to this sequence for all k > 0, and (k*2^k + 1) is the k-th Cullen number. That is, 2^A002064(k) is a term, with k > 0, from which it follows that this sequence has infinitely many terms.
Conjecture: 32 and 100 are the only terms for which x != y: 2^4 + 4^2 = 2^5 = 32 and 2^6 + 6^2 = 10^2 = 100.

			100 is a term, because 100 = 10^2 and 100 = 2^6 + 6^2.

Cf. A001597, A002064, A076980.

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A131700 Sum of all n-digit Cullen numbers.

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A264751 Triangle read by rows: T(n,k) is the number of sequences of k <= n throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the k-th throw.

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A354272 Irregular triangle read by rows: coefficients of polynomials which are the product of all possible monic Littlewood polynomials of degree n.

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A373867 Perfect powers of the form x^y + y^x, where x > 1 and y > 1.

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