A178303 -id:A178303 - cp's OEIS frontend

A331432 Triangle T(n,k) (n >= k >= 0) read by rows: T(n,0) = (1+(-1)^n)/2; for k>=1, set T(0,k) = 0, S(n,k) = binomial(n,k)*binomial(n+k+1,k), and for n>=1, T(n,k) = S(n,k)-T(n-1,k).

1, 0, 3, 1, 5, 10, 0, 10, 35, 35, 1, 14, 91, 189, 126, 0, 21, 189, 651, 924, 462, 1, 27, 351, 1749, 4026, 4290, 1716, 0, 36, 594, 4026, 13299, 22737, 19305, 6435, 1, 44, 946, 8294, 36751, 89375, 120835, 85085, 24310, 0, 55, 1430, 15730, 89375, 289003, 551837, 615043, 369512, 92378, 1, 65, 2080, 27950, 197275, 811733, 2047123, 3203837, 3031678, 1587222, 352716

Offset: 0

N. J. A. Sloane, Jan 17 2020

The scanned pages of Ser are essentially illegible, and the book is out of print and hard to locate.

For Table IV on page 93, it is simplest to ignore the minus signs. The present triangle then matches all the given terms in that triangle, so it seems best to define the triangle by the recurrences given here, and to conjecture (strongly) that this is the same as Ser's triangle.

			Triangle begins:
  1;
  0,  3;
  1,  5,   10;
  0, 10,   35,    35;
  1, 14,   91,   189,    126;
  0, 21,  189,   651,    924,    462;
  1, 27,  351,  1749,   4026,   4290,    1716;
  0, 36,  594,  4026,  13299,  22737,   19305,    6435;
  1, 44,  946,  8294,  36751,  89375,  120835,   85085,   24310;
  0, 55, 1430, 15730,  89375, 289003,  551837,  615043,  369512,   92378;
  1, 65, 2080, 27950, 197275, 811733, 2047123, 3203837, 3031678, 1587222, 352716;

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)

Columns 1 and 2 are A176222 and A331429; the last three diagonals are A002739, A002737, A001700.
Taking the component-wise sums of the rows by pairs give the triangle in A178303.
Ser's tables I and III are A331430 and A331431 (both are still mysterious).

SS := (n,k)->binomial(n,k)*binomial(n+k+1,k);
T4:=proc(n,k) local i; global SS; option remember;
if k=0 then return((1+(-1)^n)/2); fi;
if n=0 then 0 else SS(n,k)-T4(n-1,k); fi; end;
rho:=n->[seq(T4(n,k),k=0..n)];
for n from 0 to 14 do lprint(rho(n)); od:

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0, (1 + (-1)^n)/2, Binomial[n, k]*Binomial[n+k+1, k] - T[n-1, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)

def T(n,k): # A331432
    if (n<0): return 0
    elif (k==0): return ((n+1)%2)
    else: return binomial(n,k)*binomial(n+k+1,k) - T(n-1,k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

T(n, k) = binomial(n,k)*binomial(n+k+1,k) - T(n-1, k), with T(n, 0) = (1 + (-1)^n)/2.
T(n, 0) = A000035(n+1).
T(n, 1) = A176222(n).
T(n, 2) = A331429(n).
T(n, n-2) = A002739(n).
T(n, n-1) = A002737(n).
T(n, n) = A001700(n).

A178341 Smallest multiple of 3 such that decimals digits 1, ..., k (k = 1, ..., 9) and 0 appear in any order.

Original entry on oeis.org

12, 12, 123, 12234, 12345, 123456, 12234567, 12345678, 123456789, 1023456789

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), May 25 2010

			4 * 3 = 12
4 * 3 = 12
41 * 3 = 123
4078 * 3 = 12234
4115 * 3 = 12345
41152 * 3 = 123456
4078189 * 3 = 12234567
4115226 * 3 = 12345678
41152263 * 3 = 123456789
341152263 * 3 = 1023456789

Cf. A178303.

def a(n):
    digset = "".join(str(d) for d in range(1, min(n+1, 11)))
    target, lb = set(digset), int(digset) if n < 10 else 1023456789
    m = lb if lb%3 == 0 else lb + 3 - lb%3
    while target - set(str(m)): m += 3
    return m
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Dec 12 2021

A178342 Smallest prime p such that decimals digits 1, ..., k (k = 1, ..., 9) and 0 appear in any order in 3 * p.

Original entry on oeis.org

5, 7, 41, 14071, 4751, 41521, 4114189, 41115229, 411452263, 3411452263

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), May 25 2010

See A178341

Semiprime N = 3 * p, p a prime, sod(N) a multiple of 3

			5 * 3 = 15, 5 = prime(3)
7 * 3 = 21, 7 = prime(4)
41 * 3 = 123, 41 = prime(13)
14071 * 3 = 42213, 14071 = prime(1659)
4751 * 3 = 14253, 4751 = prime(640)
41521 * 3 = 124563, 41521 = prime(4343)
4114189 * 3 = 12342567, 4114189 = prime(290704)
41115229 * 3 = 123345687, 41115229 = prime(2497340)
411452263 * 3 = 1234356789, 411452263 = prime(21913969)
3411452263 * 3 = 10234356789, 3411452263 = prime(163246897)

Cf. A178303, A178341.

cp's OEIS Frontend

A331432 Triangle T(n,k) (n >= k >= 0) read by rows: T(n,0) = (1+(-1)^n)/2; for k>=1, set T(0,k) = 0, S(n,k) = binomial(n,k)*binomial(n+k+1,k), and for n>=1, T(n,k) = S(n,k)-T(n-1,k).

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A178341 Smallest multiple of 3 such that decimals digits 1, ..., k (k = 1, ..., 9) and 0 appear in any order.

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A178342 Smallest prime p such that decimals digits 1, ..., k (k = 1, ..., 9) and 0 appear in any order in 3 * p.

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