{"id":"n-e7f9db5cff35db04","revision":2,"first_index":1,"fingerprint":"f6-05007905755e7549","name":"basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","1"],["8","6","1"],["21","25","9","1"],["55","90","51","12","1"],["144","300","234","86","15","1"],["377","954","951","480","130","18","1"],["987","2939","3573","2305","855","183","21","1"],["2584","8850","12707","10008","4740","1386","245","24","1"],["6765","26195","43398","40426","23373","8715","2100","316","27","1"],["17711","76500","143682","154668","105805","48132","14770","3024","396","30","1"],["46368","221016","464148","567260","448710","241486","90342","23526","4185","485","33","1"],["121393","632916","1469778","2011260","1807585","1125036","497742","157896","35685","5610","583","36","1"],["317811","1799125","4578102","6936298","6985305","4941207","2527920","947904","260766","52030","7326","690","39","1"],["832040","5082270","14063653","23375736","26084628","20683890","12027225","5213736","1694517","411246","73425","9360","806","42","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","15","56","209","780","2911","10864","40545","151316","564719","2107560","7865521","29354524","109552575"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":"e-e9f4e7fbe6961895","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","6","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161516,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-95c725304ba970c4"]},{"id":161517,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-95c725304ba970c4","e-f4c2062c2af62bf3"]},{"id":161518,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-95c725304ba970c4","e-f4c2062c2af62bf3"]},{"id":161519,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-95c725304ba970c4","e-f4c2062c2af62bf3"]},{"id":161520,"position":4,"claim":"real_rooted","from_index":1,"region_key":"(-5, -1)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":false,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1)","region_exact":null,"edges":["e-f4c2062c2af62bf3"]}],"region_exact":{},"incoming_edges":[{"id":"e-95c725304ba970c4","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-e7f9db5cff35db04","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a8b150c3ffce36e03b3a08d0d6e89e014c973b17fe5a28f6b0d5f2fd088b6f6a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-976cdb6e4a693ec8","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-e9f4e7fbe6961895","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-e7f9db5cff35db04","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"c8acf349aeb26f88aceb897d4f78289afd2a334832fc6f6495b5beb63ae276eb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-f4c2062c2af62bf3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-e7f9db5cff35db04","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"02b7eb9f98602147e7f72b3adcd65804252a5aa8637ad015afe48995abd7c5cb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":false,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-976cdb6e4a693ec8","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-7df9a7fd3f4f500b","kind":"fingerprint_related","node_a_id":"n-e7f9db5cff35db04","node_b_id":"n-46fcf1ae8116c282","fingerprint":"f6-05007905755e7549","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-05007905755e7549"},"other_id":"n-46fcf1ae8116c282","other":{"id":"n-46fcf1ae8116c282","name":"mobius_x_over_x_plus_one of Even row Pascal-square read by antidiagonals.","first_index":0,"fingerprint":"f6-05007905755e7549","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","6","8"]],"card_rows":[["1"],["1","3"],["1","6","8"],["1","9","25","21"]],"row_sums_preview":["1","4","15","56","209","780","2911","10864"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79ebafb266da3b41","steps":["x \u21a6 x/(x+1)"],"edges":["e-83ce870cdc6237f7"],"text":"A172431 \u2192 x \u21a6 x/(x+1)"},"headline":"A172431 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-94fa355a4c9eb5d2","kind":"fingerprint_related","node_a_id":"n-e7f9db5cff35db04","node_b_id":"n-c4e1c7f7d91f2e30","fingerprint":"f6-05007905755e7549","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-05007905755e7549"},"other_id":"n-c4e1c7f7d91f2e30","other":{"id":"n-c4e1c7f7d91f2e30","name":"reversal of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","first_index":0,"fingerprint":"f6-05007905755e7549","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","6","8"]],"card_rows":[["1"],["1","3"],["1","6","8"],["1","9","25","21"]],"row_sums_preview":["1","4","15","56","209","780","2911","10864"],"regions":["(-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f287bed89e07609f","steps":["reverse"],"edges":["e-a162795980dd8c54"],"text":"A125662 \u2192 reverse"},"headline":"A125662 \u2192 reverse","chain":null}},{"id":"l-e5d24b1b264dc890","kind":"fingerprint_related","node_a_id":"n-f287bed89e07609f","node_b_id":"n-e7f9db5cff35db04","fingerprint":"f6-05007905755e7549","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-05007905755e7549"},"other_id":"n-f287bed89e07609f","other":{"id":"n-f287bed89e07609f","name":"A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","first_index":0,"fingerprint":"f6-05007905755e7549","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["8","6","1"]],"card_rows":[["1"],["3","1"],["8","6","1"],["21","25","9","1"]],"row_sums_preview":["1","4","15","56","209","780","2911","10864"],"regions":["(-inf, +inf)","(-inf, 0)","(-5, -1)"],"region_exact":{},"oeis_ids":["A125662"],"derivation":{"root":"n-f287bed89e07609f","steps":[],"edges":[],"text":"A125662"},"headline":"A125662: A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","chain":null}},{"id":"l-2405b1b210fe0c12","kind":"oeis_match","node_a_id":"n-e7f9db5cff35db04","node_b_id":null,"fingerprint":null,"oeis_id":"A001353","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001353","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-250385104b16a204","kind":"oeis_match","node_a_id":"n-e7f9db5cff35db04","node_b_id":null,"fingerprint":null,"oeis_id":"A010905","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A010905","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7d5fc88848349261","kind":"oeis_match","node_a_id":"n-e7f9db5cff35db04","node_b_id":null,"fingerprint":null,"oeis_id":"A195503","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A195503","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fd492049b0729615","kind":"oeis_match","node_a_id":"n-e7f9db5cff35db04","node_b_id":null,"fingerprint":null,"oeis_id":"A125662","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A125662","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001353","relation":"match_row_sums"},{"oeis_id":"A010905","relation":"match_row_sums"},{"oeis_id":"A195503","relation":"match_row_sums"},{"oeis_id":"A125662","relation":"match_triangle"}],"bfiles":[{"node_id":"n-e7f9db5cff35db04","oeis_id":"A010905","target":"row_sums","oeis_offset":0,"first_row":2,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":12490},{"node_id":"n-e7f9db5cff35db04","oeis_id":"A125662","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":16894},{"node_id":"n-e7f9db5cff35db04","oeis_id":"A195503","target":"row_sums","oeis_offset":1,"first_row":2,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":12492}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    3,     1;\n   3:    8,     6,     1;\n   4:   21,    25,     9,     1;\n   5:   55,    90,    51,    12,     1;\n   6:  144,   300,   234,    86,    15,    1;\n   7:  377,   954,   951,   480,   130,   18,    1;\n   8:  987,  2939,  3573,  2305,   855,  183,   21,   1;\n   9: 2584,  8850, 12707, 10008,  4740, 1386,  245,  24,  1;\n  10: 6765, 26195, 43398, 40426, 23373, 8715, 2100, 316, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (3 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A125662","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    3,     1;\n  2:    8,     6,     1;\n  3:   21,    25,     9,     1;\n  4:   55,    90,    51,    12,     1;\n  5:  144,   300,   234,    86,    15,    1;\n  6:  377,   954,   951,   480,   130,   18,    1;\n  7:  987,  2939,  3573,  2305,   855,  183,   21,   1;\n  8: 2584,  8850, 12707, 10008,  4740, 1386,  245,  24,  1;\n  9: 6765, 26195, 43398, 40426, 23373, 8715, 2100, 316, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (3 + x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["basis A007318"],"edges":["e-95c725304ba970c4"],"text":"A078812 \u2192 basis A007318"},"headline":"A078812 \u2192 basis A007318","chain":null}
