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Also called the Catalan triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-21212c0e506cddce","kind":"fingerprint_related","node_a_id":"n-0fdfc788c35d0f4b","node_b_id":"n-7ddb81b454012205","fingerprint":"f6-db52feb71393f616","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-db52feb71393f616"},"other_id":"n-7ddb81b454012205","other":{"id":"n-7ddb81b454012205","name":"theta_plus_one_squared of basis_map_A013610 of narayana","first_index":1,"fingerprint":"f6-db52feb71393f616","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","12"],["5","60","81"]],"card_rows":[["1"],["2","12"],["5","60","81"],["14","252","729","432"]],"row_sums_preview":["1","14","146","1427","13659","129480","1220100","11447525"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-21b441a6a6a148f7","e-71437b7db73c9430"],"text":"narayana \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"narayana \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}}],"oeis":[],"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:     1;\n  1:     2,     12;\n  2:     5,     60,       81;\n  3:    14,    252,      729,      432;\n  4:    42,   1008,     4536,     6048,      2025;\n  5:   132,   3960,    24300,    51840,     40500,      8748;\n  6:   429,  15444,   120285,   356400,    455625,    236196,     35721;\n  7:  1430,  60060,   567567,  2162160,   3898125,   3367980,   1250235,    139968;\n  8:  4862, 233376,  2594592, 12108096,  28378350,  35026992,  22004136,   6158592,   531441;\n  9: 16796, 906984, 11599848, 64157184, 185749200, 300913704, 273051324, 131010048, 28697814, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+16*n^2+20*n+8+(18*n^4+63*n^3+57*n^2-12)*x+(9*n^5+36*n^4+45*n^3+18*n^2)*x^2)*P(n,x) = (16*n^3+40*n^2+32*n+8+(72*n^4+288*n^3+354*n^2+126*n-12)*x+(36*n^5+360*n^4+810*n^3+522*n^2)*x^2+(81*n^5+378*n^4+540*n^3+243*n^2+54*n)*x^3)*P(n-1,x) - ((8*n^2+16*n+8)*x+(36*n^3+132*n^2+141*n+42)*x^2+(18*n^4+198*n^3+468*n^2+360*n+72)*x^3+(54*n^4+270*n^3+432*n^2+243*n+54)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 40*n^2 + 32*n + 8 + (72*n^4 + 288*n^3 + 354*n^2 + 126*n - 12)*x + (36*n^5 + 360*n^4 + 810*n^3 + 522*n^2)*x^2 + (81*n^5 + 378*n^4 + 540*n^3 + 243*n^2 + 54*n)*x^3)*P[n - 1, x] - ((8*n^2 + 16*n + 8)*x + (36*n^3 + 132*n^2 + 141*n + 42)*x^2 + (18*n^4 + 198*n^3 + 468*n^2 + 360*n + 72)*x^3 + (54*n^4 + 270*n^3 + 432*n^2 + 243*n + 54)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 16*n^2 + 20*n + 8 + (18*n^4 + 63*n^3 + 57*n^2 - 12)*x + (9*n^5 + 36*n^4 + 45*n^3 + 18*n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 40*n^2 + 32*n + 8 + (72*n^4 + 288*n^3 + 354*n^2 + 126*n - 12)*x + (36*n^5 + 360*n^4 + 810*n^3 + 522*n^2)*x^2 + (81*n^5 + 378*n^4 + 540*n^3 + 243*n^2 + 54*n)*x^3)*v[i - 1] - ((8*n^2 + 16*n + 8)*x + (36*n^3 + 132*n^2 + 141*n + 42)*x^2 + (18*n^4 + 198*n^3 + 468*n^2 + 360*n + 72)*x^3 + (54*n^4 + 270*n^3 + 432*n^2 + 243*n + 54)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 16*n^2 + 20*n + 8 + (18*n^4 + 63*n^3 + 57*n^2 - 12)*x + (9*n^5 + 36*n^4 + 45*n^3 + 18*n^2)*x^2)); 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-c2eaf5212a4dec13","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-9081e2b3ba0d880f","e-3fc5d9665945878f"],"text":"A001263 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A001263 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}