A053445 -id:A053445 - cp's OEIS frontend

A378970 Antidiagonal-sums of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

1, 1, 1, 5, -4, 18, -20, 47, -56, 110, -153, 309, -532, 1045, -1768, 2855, -3620, 2928, 2927, -20371, 62261, -148774, 314112, -613835, 1155936, -2175658, 4244218, -8753316, 19006746, -42471491, 95234915, -210395017, 453414314, -949507878, 1931940045

Offset: 0

Gus Wiseman, Dec 14 2024

sign

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = -4.

For primes we have A140119 or A376683, absolute value A376681 or A376684.
For composites we have A377034, absolute value A377035.
For squarefree numbers we have A377039, absolute value A377040.
For nonsquarefree numbers we have A377047, absolute value A377048.
For prime powers we have A377052, absolute value A377053.
For partition numbers we have A377056, absolute value A378621.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The unsigned version is A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244.

nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A059797 Second in a series of arrays counting standard tableaux by partition type.

Original entry on oeis.org

2, 5, 5, 9, 16, 9, 14, 35, 35, 14, 20, 64, 90, 64, 20, 27, 105, 189, 189, 105, 27, 35, 160, 350, 448, 350, 160, 35, 44, 231, 594, 924, 924, 594, 231, 44, 54, 320, 945, 1728, 2100, 1728, 945, 320, 54, 65, 429, 1430, 3003, 4290, 4290, 3003, 1430, 429, 65

Offset: 0

Alford Arnold, Feb 22 2001

The first array in the series is Pascal's triangle, A007318. The initial partition for each subsequent array in the series is chosen as described in A053445. When cells are squared, as in A008459, row sums yield 1, 2, 6, 24, ... (A000142). E.g., (1 + 16 + 36 + 16 + 1) + (25 + 25) = 70 + 50 = 120 using row five from A007318 and row two from this array.

Number of lattice paths from (0,0) to (n,k) in exactly n+k+2 steps. Moves allowed: (0,1), (0,-1), (-1,0) and (1,0). Paths must stay in Quadrant I but may touch the axes. - Juan Luis Vargas-Molina, Mar 03 2014

			a(5) = 16 because we can write T(2,2) = T(1,2) + T(1,1) + A007318(4,2) = 5 + 5 + 6.
   2;
   5,   5;
   9,  16,   9;
  14,  35,  35,  14;
  20,  64,  90,  64,  20;
  27, 105, 189, 189, 105,  27;
T(n,k) as a grid for the first few lattice points. Where T(0,0)=2 since there are two ways to get to (0,0) with "0+0+2" steps under the move restrictions.
     k = 0  1   2   3    4    5
T(0,k) = 2  5   9   14   20   27
T(1,k) = 5  16  35  64   105  160
T(2,k) = 9  35  90  189  350  594
T(3,k) = 14 64  189 448  924  1728
T(4,k) = 20 105 350 924  2100 4290
T(5,k) = 27 160 594 1728 4290 9504
- _Juan Luis Vargas-Molina_, Mar 03 2014

Stanton and White, Constructive Combinatorics, 1986, pp. 84, 91.

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.
Juan Luis Vargas-Molina, C++ Algorithm

Cf. A000041, A000142, A007318, A008459, A053445.

A059797 := proc(n,k) option remember; if n <0 or k<0 or k>n then 0; else procname(n-1,k-1)+procname(n-1,k)+binomial(n+2,k+1) ; end if; end proc:
seq(seq( A059797(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Jan 27 2011

t[n_, k_] /; (n < 0 || k < 0 || k > n) = 0; t[n_, k_] := t[n, k] = t[n - 1, k - 1] + t[n - 1, k] + Binomial[n + 2, k + 1]; Flatten[ Table[ t[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Dec 20 2011, after R. J. Mathar *)
T[n_, k_] := (k + 1)(n + k +4) FactorialPower[k + n + 2, n]/(n! + (n + 1)!)
MatrixForm[Table[T[n, k], {n, 0, 10}, {k, 0, 10}]] (* Juan Luis Vargas-Molina, Mar 03 2014 *)

T(row, col) = T(row-1, col-1) + T(row-1, col) + A007318(row+2, col+1). - Corrected by R. J. Mathar, Jan 27 2011

T(n,k) = (k+1)(n+k+4)FallingFactorial(n+k+2,n)/((n+1)!+n!) n,k >= 0. - Juan Luis Vargas-Molina, Mar 03 2014

A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

Original entry on oeis.org

1, 1, 1, 5, 8, 18, 30, 47, 70, 110, 177, 309, 574, 1063, 1892, 3107, 4598, 6166, 8737, 20603, 62457, 149132, 314116, 614093, 1155968, 2176048, 4244322, 8753864, 19006756, 42472117, 95235017, 210396059, 453414950, 949510166, 1931941261, 3826650257, 7400745917

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = 8.

For primes we have A376681 or A376684, signed version A140119 or A376683.
For composites we have A377035, signed version A377034.
For squarefree numbers we have A377040, signed version A377039.
For nonsquarefree numbers we have A377048, signed version A377049.
For prime powers we have A377053, signed version A377052.
For partition numbers we have A378621, signed version A377056.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The signed version is A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244, A325245, A325268.

nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 5, 11, 16, 36, 65, 142, 285, 595, 1210, 2497, 5134, 10726, 22637, 48383, 104066, 224296, 481985, 1030299, 2188912, 4626313, 9743750, 20492711, 43114180, 90843475, 191776658, 405528200, 858384333, 1817311451, 3845500427, 8129033837, 17162815092

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 5.

These are the antidiagonal-sums of the absolute value of A175804.
First column of the same array is A281425.
For primes we have A376681 or A376684, signed A140119 or A376683.
For composites we have A377035, signed A377034.
For squarefree numbers we have A377040, signed A377039.
For nonsquarefree numbers we have A377048, signed A377049.
For prime powers we have A377053, signed A377052.
The signed version is A377056.
The corresponding array for strict partitions is A378622, see A293467, A377285, A378971, A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A098859, A116608, A225486, A325242, A325244, A325268.

nn=30;
q=Table[PartitionsP[n],{n,0,nn}];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}]
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A138533 Resort the multinomial sequence A036038 by source partition as described in A126442, A129306 and A136101.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 6, 120, 60, 20, 5, 1, 30, 10, 720, 360, 120, 30, 6, 1, 180, 60, 15, 20, 90

Offset: 1

Alford Arnold, Mar 27 2008

Multinomials count permutations of multisets and also paths in lattices; for example, there are six paths (from null to full) through the lattice of divisors for signature 36: 2233 2323 2332 3223 3232 and 3322.

			a(11) is six because the eleventh least prime signature in source format is 36 the signature for partition 2+2 the ninth partition and A036038(9) = 6.
The tables begin:
1.......2.......6.......24......120.....720....5040.....40320......362880
........1.......3.......12.......60.....360....2520.....20160......181440
................1.......4........20.....120.....840......6720.......60480
........................1........5.......30.....210......1680.......15120
.. ..............................1........6......42......336........3024
..........................................1.......7.......56.........504
..................................................1........8..........72
...........................................................1...........9
.......................................................................1
........................6........30.....180....1260....10080........90720
.................................10......60.....420.....3360........30240
...

Cf. A000142 A001710 A005651 (column sums) A025487 A053445.

Cf. A173333. [From Reinhard Zumkeller, Feb 19 2010]

A119712 a(n) is the smallest integer k such that the n-th (forward) difference of the partition sequence A000041 is positive from k onwards.

Original entry on oeis.org

0, 1, 6, 23, 64, 129, 222, 345, 502, 695, 924, 1193, 1502, 1853, 2246, 2687, 3172, 3705, 4286, 4917, 5600, 6333, 7118, 7957, 8848, 9797, 10800, 11861, 12978, 14153, 15386, 16681, 18034, 19447, 20922, 22459, 24060, 25723, 27448, 29239, 31094, 33015

Offset: 0

Moshe Shmuel Newman, Jun 11 2006

nonn

The first entry is considered to be indexed by zero. For example, the third difference A072380 starts with -1,1 and continues alternating in sign till the 24th entry, from which point it is positive.

Using a different (backward) definition of the difference operator, this sequence has also been given as 1, 8, 26, 68, 134, 228, 352, ... A155861.

Jean-François Alcover, Table of n, a(n) for n = 0..60
Gert Almkvist, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.
I. J. Good, Problem 6137, American Mathematical Monthly, 1978, pages 830-831.
Hansraj Gupta, Finite Differences of the Partition Function, Math. Comp. 32 (1978), 1241-1243.
A. M. Odlyzko, Differences of the partition function, Acta Arithmetica 49.3 (1988): 237-254.
Eric Weisstein's World of Mathematics, Forward Difference.

Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A155861.

with(combinat): DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end: a:= proc(n) option remember; local f, k; if n=0 then 0 else f:= (DD@@n)(numbpart); for k from a(n-1) while not (f(k)>0 and f(k+1)>0) do od; k fi end: seq(a(n), n=0..20); # Alois P. Heinz, Jul 20 2009

a[n_] := a[n] = Module[{f}, f[i_] = DifferenceDelta[PartitionsP[i], {i, n}]; For[j = 2, True, j++, If[f[j] > 0 && f[j+1] > 0, Return[j]]]];
a[0] = 0; a[1] = 1;
Table[Print[n, " ", a[n]]; a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 04 2020 *)

Odlyzko gives an asymptotic formula a(n)~(6/(Pi)^2) * (n log n)^2

a(11)-a(41) from Alois P. Heinz, Jul 20 2009

A155861 a(n) is the smallest integer k such that the n-th (backward) difference of the partition sequence A000041 is positive from k onwards.

Original entry on oeis.org

1, 2, 8, 26, 68, 134, 228, 352, 510, 704, 934, 1204, 1514, 1866, 2260, 2702, 3188, 3722, 4304, 4936, 5620, 6354, 7140, 7980, 8872, 9822, 10826, 11888, 13006, 14182, 15416, 16712, 18066, 19480, 20956, 22494, 24096, 25760, 27486, 29278, 31134

Offset: 0

Alois P. Heinz, Dec 16 2010

nonn

Using a different (forward) definition of the difference operator, this sequence has also been given as 0, 1, 6, 23, 64, 129, 222, ... A119712.

Jean-François Alcover, Table of n, a(n) for n = 0..60
Gert Almkvist, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.
Hansraj Gupta, Finite Differences of the Partition Function, Math. Comp. 32 (1978), 1241-1243.
Charles Knessl, Asymptotic Behavior of High-Order Differences of the Partition Function, Communications on Pure and Applied Mathematics, 44 (1991), 1033-1045.
A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), 237-254.
Eric Weisstein's World of Mathematics, Backward Difference

Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A119712.

A41:= n-> `if` (n<0, 0, combinat[numbpart](n)):
DB:= proc(p)
       proc(n) option remember;
         p(n) -p(n-1)
       end
     end:
a:= proc(n) option remember;
      local f, k;
      if n=0 then 1
             else f:= (DB@@n)(A41);
             for k from a(n-1) while not (f(k)>0 and f(k+1)>0) do od; k
      fi
    end:
seq(a(n), n=0..20);

a[n_] := a[n] = Module[{f}, f[i_] = DifferenceDelta[PartitionsP[i], {i, n}]; For[j = 2, True, j++, If[f[j] > 0 && f[j + 1] > 0, Return[j + n]]]];
a[0] = 1; a[1] = 2;
Table[Print[n, " ", a[n]]; a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 04 2020 *)

An asymptotic formula is a(n) ~ 6/Pi^2 * n^2 (log n)^2.

A091307 a(n)=6*2^n+4 (Bode Number A003461(n+2)) except for a(1)=6.

Original entry on oeis.org

6, 28, 52, 100, 196, 388, 772, 1540, 3076, 6148, 12292, 24580, 49156, 98308, 196612, 393220, 786436, 1572868, 3145732, 6291460, 12582916, 25165828, 50331652, 100663300, 201326596, 402653188, 805306372, 1610612740, 3221225476, 6442450948, 12884901892

Offset: 1

Alford Arnold, Feb 21 2004

Sequence similar to Bode Numbers relevant to A079946 and numeric partitions.
A053445 describes certain partitions which start triangular arrays of all other numeric partitions; e.g. - 22, 33, 222, 44, 332, 2222, ... A079946 provides the indices for these partitions. (cf. A090324 and A090774).
By expanding the terms of a(n) in a similar manner, the vertex partitions can be readily indexed by noting that the indices increase by eight as follows: 6 28 (one case), 52 60 (two cases), 100 108 116 124 (four cases), 196 204 212 220 228 236 244 252 (eight cases), 388 ...

			a(3) = 52 because we can write 52 = 2*28 - 4.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Except for initial term, same as A003461(n+2). Cf. A053445, A079946, A090774.

CoefficientList[Series[2x (3+5x-10x^2)/((1-x)(1-2x)),{x,0,30}],x] (* or *) LinearRecurrence[{3,-2},{0,6,28,52},40] (* Harvey P. Dale, Sep 01 2021 *)

a(1) = 6, a(2) = 28, a(n) = 2*a(n-1) - 4 for n > 2.

G.f.: 2*x*(3+5*x-10*x^2)/((1-x)*(1-2*x)). - Colin Barker, Mar 12 2012

Edited by M. F. Hasler, Apr 07 2009

A160646 First differences of A160644.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 4, 8, 10, 13, 22, 28, 39, 58, 77, 104, 148, 197, 265, 363, 481, 638, 858, 1126, 1480, 1953, 2544, 3309, 4312, 5566, 7175, 9246, 11843, 15136, 19328, 24564, 31158, 39466, 49811, 62737, 78900, 98931, 123817, 154707, 192830, 239911, 298013

Offset: 1

Alford Arnold, May 24 2009

Second differences count a subset of unrestricted partitions; cf. A160648.

			A160644 begins 1, 1, 2, 3, 4, 7, 10, 14, 22, 32, 45, 67, 95, 134, 192, ... so a(n) begins 0, 1, 1, 1, 3, 3, 4, 8, 10, 13, 22, 28, 39, 58, ...

Nathaniel Johnston, Table of n, a(n) for n = 1..2500

Cf. A000041, A002685, A053445, A160644, A160648.

A160648 Second differences of sequence A160644.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 1, 4, 2, 3, 9, 6, 11, 19, 19, 27, 44, 49, 68, 98, 118, 157, 220, 268, 354, 473, 591, 765, 1003, 1254, 1609, 2071, 2597, 3293, 4192, 5236, 6594, 8308, 10345, 12926, 16163, 20031, 24886, 30890, 38123, 47081, 58102, 71381, 87704, 107643

Offset: 1

Alford Arnold, May 25 2009

A160644 bisects sequence A053445 which counts unrestricted partitions such that the two largest values match and that no part is less than two.
Conjecture: a(n) counts unrestricted partitions of even numbers such that
the three largest values match and that, after "222", no part is less than three.

			a(n) begins 0 1 0 0 2 0 1 4 2 3 9 ... and counts 222; 444,3333;
666,5553,444433,333333; 5555,44444; 6664,55543,444433;
888,6666,7773,55554,66633,444444,555333,4443333,33333333; ...

Nathaniel Johnston, Table of n, a(n) for n = 1..2500

Cf. A000041, A002865, A053445, A160644, A160646.

cp's OEIS Frontend

A378970 Antidiagonal-sums of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

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A059797 Second in a series of arrays counting standard tableaux by partition type.

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A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

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A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

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A138533 Resort the multinomial sequence A036038 by source partition as described in A126442, A129306 and A136101.

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A119712 a(n) is the smallest integer k such that the n-th (forward) difference of the partition sequence A000041 is positive from k onwards.

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A155861 a(n) is the smallest integer k such that the n-th (backward) difference of the partition sequence A000041 is positive from k onwards.

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A091307 a(n)=6*2^n+4 (Bode Number A003461(n+2)) except for a(1)=6.

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