{"id":"n-e4013d25fc6715f3","revision":1,"first_index":0,"fingerprint":"f6-12f8b318db9d845e","name":"gamma_polynomial of Pascal-(1,8,1) array.","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","8"],["1","16"],["1","24","64"],["1","32","192"],["1","40","384","512"],["1","48","640","2048"],["1","56","960","5120","4096"],["1","64","1344","10240","20480"],["1","72","1792","17920","61440","32768"],["1","80","2304","28672","143360","196608"],["1","88","2880","43008","286720","688128","262144"],["1","96","3520","61440","516096","1835008","1835008"],["1","104","4224","84480","860160","4128768","7340032","2097152"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","9","17","89","225","937","2737","10233","32129","113993","371025","1282969","4251169","14514921"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162504,"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-023010f9acf2a31f"]},{"id":162505,"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-023010f9acf2a31f"]},{"id":162506,"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-023010f9acf2a31f"]}],"region_exact":{},"incoming_edges":[{"id":"e-023010f9acf2a31f","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-e4013d25fc6715f3","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":"342bf13768780537b99ffb6b85f720622a62d3b9b08c7118e93e9041c444e3aa","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-1c472d1762258215","role":"sequence","name":"Pascal-(1,8,1) array.","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-1f9cb2c869a29c04","kind":"oeis_match","node_a_id":"n-e4013d25fc6715f3","node_b_id":null,"fingerprint":null,"oeis_id":"A317026","in_catalog":0,"target":"triangle","row_alignment":"node_lengths","rows_matched":14,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":14,"target":"triangle","terms_compared":56},"in_catalog":false,"oeis_id":"A317026","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ba0639ae54a18ec9","kind":"oeis_match","node_a_id":"n-e4013d25fc6715f3","node_b_id":null,"fingerprint":null,"oeis_id":"A015443","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A015443","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A015443","relation":"match_row_sums"},{"oeis_id":"A317026","relation":"match_triangle"}],"bfiles":[{"node_id":"n-e4013d25fc6715f3","oeis_id":"A015443","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":11451},{"node_id":"n-e4013d25fc6715f3","oeis_id":"A317026","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":62,"terms_matched":56,"oeis_terms_agreeing":56,"oeis_terms_available":56,"term_count":992,"raw_size":19513}],"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,  8;\n  3: 1, 16;\n  4: 1, 24,   64;\n  5: 1, 32,  192;\n  6: 1, 40,  384,   512;\n  7: 1, 48,  640,  2048;\n  8: 1, 56,  960,  5120,  4096;\n  9: 1, 64, 1344, 10240, 20480;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-1,x) + 8*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 8*x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + 8*x*v[i - 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}},{"oeis_id":"A317026","target":"triangle","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,  8;\n  3: 1, 16;\n  4: 1, 24,   64;\n  5: 1, 32,  192;\n  6: 1, 40,  384,   512;\n  7: 1, 48,  640,  2048;\n  8: 1, 56,  960,  5120,  4096;\n  9: 1, 64, 1344, 10240, 20480;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-1,x) + 8*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 8*x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + 8*x*v[i - 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-1c472d1762258215","steps":["\u03b3"],"edges":["e-023010f9acf2a31f"],"text":"A143683 \u2192 \u03b3"},"headline":"A143683 \u2192 \u03b3","chain":null}