{"id":"n-4749196d174a5d58","revision":1,"first_index":0,"fingerprint":"f6-4924f2415393e159","name":"gamma_polynomial of Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","6"],["1","36"],["1","162","216"],["1","672","3456"],["1","2718","36288","20736"],["1","10908","319248","642816"],["1","43674","2561976","12174624","3856896"],["1","174744","19506096","183233664","196235136"],["1","699030","143881920","2417519520","5849262720","1177410816"],["1","2796180","1040726880","29354944320","134060883840","89029725696"],["1","11184786","7436081880","337259972160","2623439819520","3837571520256","534178354176"],["1","44739216","52723660320","3729531651840","46246076651520","124363699992576","56200246972416"],["1","178956942","372018410496","40142394175680","757839325847040","3377201499426816","3306351292342272","337201481834496"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","37","379","4129","59743","972973","18637171","399149641","9588774007","253489076917","7339896932779","230592323667889","7819108190993743"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-6","0"],["0","6","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-12"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162582,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-5a037b5991384369"]},{"id":162583,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-5a037b5991384369"]},{"id":162584,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-5a037b5991384369"]}],"region_exact":{},"incoming_edges":[{"id":"e-5a037b5991384369","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-4749196d174a5d58","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"5082f0ccbcae88620508f596451c57247a2b426554dc277e74d8d29ba3267c16","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ddb0cd3a4d6edf09","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).","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-f805f8b7b7994426","kind":"fingerprint_related","node_a_id":"n-4749196d174a5d58","node_b_id":"n-86b59eb5d2c0ef87","fingerprint":"f6-4924f2415393e159","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4924f2415393e159"},"other_id":"n-86b59eb5d2c0ef87","other":{"id":"n-86b59eb5d2c0ef87","name":"gamma_polynomial of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.","first_index":0,"fingerprint":"f6-4924f2415393e159","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","24"]],"card_rows":[["1"],["2"],["4","24"],["8","288"]],"row_sums_preview":["1","2","28","296","6064","132128","3823552","124540544"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c060a08f0b5770ad","steps":["\u03b3"],"edges":["e-adb175137addd81e"],"text":"A257616 \u2192 \u03b3"},"headline":"A257616 \u2192 \u03b3","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: 1;\n  2: 1,      6;\n  3: 1,     36;\n  4: 1,    162,      216;\n  5: 1,    672,     3456;\n  6: 1,   2718,    36288,     20736;\n  7: 1,  10908,   319248,    642816;\n  8: 1,  43674,  2561976,  12174624,   3856896;\n  9: 1, 174744, 19506096, 183233664, 196235136;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(6*n-6)*x)*P(n-1,x) + (3*x-12*x^2)*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[(1 + (6*n - 6)*x)*P[n - 1, x] + (3*x - 12*x^2)*D[P[n - 1, x], x]];\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] = (1 + (6*n - 6)*x)*v[i - 1] + (3*x - 12*x^2)*deriv(v[i - 1], x)); 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-ddb0cd3a4d6edf09","steps":["\u03b3"],"edges":["e-5a037b5991384369"],"text":"A142458 \u2192 \u03b3"},"headline":"A142458 \u2192 \u03b3","chain":null}
