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A347056 Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.

%I A347056 #13 May 29 2022 21:43:14
%S A347056 1,3,4,6,16,10,10,40,50,20,15,80,150,120,35,21,140,350,420,245,56,28,
%T A347056 224,700,1120,980,448,84,36,336,1260,2520,2940,2016,756,120,45,480,
%U A347056 2100,5040,7350,6720,3780,1200,165,55,660,3300,9240,16170,18480,13860,6600,1815,220
%N A347056 Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.
%C A347056 This triangle is T[3] in the sequence (T[p]) of triangles defined by: T[p](n,k) = (k+p)*(n+p-1)!/(k!*(n-k)!*p!) and T[0](0,0)=1.
%C A347056 Riordan triangle (1/(1-x)^3, x/(1-x)) with column k scaled with A000292(k+1) = binomial(k+3, 3), for k >= 0. - _Wolfdieter Lang_, Sep 30 2021
%H A347056 Luc Rousseau, <a href="/A347056/a347056.svg">Illustration: the A347056 triangle in a sequence of contiguous triangles.</a>
%F A347056 T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6.
%F A347056 G.f. column k: x^k*binomial(k+3, 3)/(1 - x)^(k+3), for k >= 0. - _Wolfdieter Lang_, Sep 30 2021
%e A347056 T(6,2) = (6+1)*(6+2)*(2+3)*binomial(6,2)/6 = 7*8*5*15/6 = 700.
%e A347056 The triangle T begins:
%e A347056 n \ k  0   1    2     3     4     5     6     7     8    9  10 ...
%e A347056 0:     1
%e A347056 1:     3   4
%e A347056 2:     6  16   10
%e A347056 3:    10  40   50    20
%e A347056 4:    15  80  150   120    35
%e A347056 5:    21 140  350   420   245    56
%e A347056 6:    28 224  700  1120   980   448    84
%e A347056 7:    36 336 1260  2520  2940  2016   756   120
%e A347056 8:    45 480 2100  5040  7350  6720  3780  1200   165
%e A347056 9:    55 660 3300  9240 16170 18480 13860  6600  1815  220
%e A347056 10:   66 880 4950 15840 32340 44352 41580 26400 10890 2640 286
%e A347056 ... - _Wolfdieter Lang_, Sep 30 2021
%o A347056 (PARI)
%o A347056 T(p,n,k)=if(n==0&&p==0,1,((k+p)*(n+p-1)!)/(k!*(n-k)!*p!))
%o A347056 for(n=0,9,for(k=0,n,print1(T(3,n,k),", ")))
%Y A347056 Cf. A097805 (p=0), A103406 (p=1), A124932 (essentially p=2).
%Y A347056 From _Wolfdieter Lang_, Sep 30 2021: (Start)
%Y A347056 Columns (with leading zeros): A000217(n+1), 4*A000294, 10*A000332(n+2), 20*A000389(n+2), 35*A000579(n+2), 56*A000580(n+2), 84*A000581(n+2), 120*A000582(n+2), ...
%Y A347056 Diagonals: A000292(k+1), A004320(k+1), 2*A006411(k+1), 10*A040977, ... (End)
%K A347056 nonn,tabl,easy
%O A347056 0,2
%A A347056 _Luc Rousseau_, Aug 14 2021