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Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","chain":null}},{"id":"l-fd1b638b65f57389","kind":"fingerprint_related","node_a_id":"n-97228a88c81561dc","node_b_id":"n-fe49f5f39873300f","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-97228a88c81561dc","other":{"id":"n-97228a88c81561dc","name":"Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","6"],["21","42","21"]],"card_rows":[["1"],["6","6"],["21","42","21"],["56","168","168","56"]],"row_sums_preview":["1","12","84","448","2016","8064","29568","101376"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A178822"],"derivation":{"root":"n-97228a88c81561dc","steps":[],"edges":[],"text":"A178822"},"headline":"A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","chain":null}},{"id":"l-2d5470b9d4daebdb","kind":"oeis_match","node_a_id":"n-fe49f5f39873300f","node_b_id":null,"fingerprint":null,"oeis_id":"A009992","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A009992","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009992","relation":"match_row_sums"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:              576,              1152,               576;\n  1:            13824,             41472,             41472,              13824;\n  2:           331776,           1327104,           1990656,            1327104,             331776;\n  3:          7962624,          39813120,          79626240,           79626240,           39813120,            7962624;\n  4:        191102976,        1146617856,        2866544640,         3822059520,         2866544640,         1146617856,          191102976;\n  5:       4586471424,       32105299968,       96315899904,       160526499840,       160526499840,        96315899904,        32105299968,         4586471424;\n  6:     110075314176,      880602513408,     3082108796928,      6164217593856,      7705271992320,      6164217593856,      3082108796928,       880602513408,       110075314176;\n  7:    2641807540224,    23776267862016,    95105071448064,    221911833378816,    332867750068224,    332867750068224,    221911833378816,     95105071448064,     23776267862016,     2641807540224;\n  8:   63403380965376,   634033809653760,  2853152143441920,   7608405715845120,  13314710002728960,  15977652003274752,  13314710002728960,   7608405715845120,   2853152143441920,   634033809653760,    63403380965376;\n  9: 1521681143169024, 16738492574859264, 83692462874296320, 251077388622888960, 502154777245777920, 703016688144089088, 703016688144089088, 502154777245777920, 251077388622888960, 83692462874296320, 16738492574859264, 1521681143169024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (24+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 1152*x + 576*x^2.","mathematica":"P[0, x_] := 576 + 1152*x + 576*x^2;\nP[n_, x_] := P[n, x] = Expand[(24 + 24*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 576 + 1152*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 24*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013609"],"edges":["e-9ec880266e74c89d"],"text":"A038338 \u2192 basis A013609"},"headline":"A038338 \u2192 basis A013609","chain":null}