{"id":"n-05c39cd3844b146d","revision":2,"first_index":0,"fingerprint":"f6-b1029f17d1b64336","name":"basis_map_A007318 of Triangle of coefficients in expansion of (2+3x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["5","3"],["25","30","9"],["125","225","135","27"],["625","1500","1350","540","81"],["3125","9375","11250","6750","2025","243"],["15625","56250","84375","67500","30375","7290","729"],["78125","328125","590625","590625","354375","127575","25515","2187"],["390625","1875000","3937500","4725000","3543750","1701000","510300","87480","6561"],["1953125","10546875","25312500","35437500","31893750","19136250","7654500","1968300","295245","19683"],["9765625","58593750","158203125","253125000","265781250","191362500","95681250","32805000","7381125","984150","59049"],["48828125","322265625","966796875","1740234375","2088281250","1754156250","1052493750","451068750","135320625","27064125","3247695","177147"],["244140625","1757812500","5800781250","11601562500","15662109375","15035625000","10524937500","5412825000","2029809375","541282500","97430850","10628820","531441"],["1220703125","9521484375","34277343750","75410156250","113115234375","122164453125","97731562500","58638937500","26387521875","8795840625","2111001750","345436650","34543665","1594323"],["6103515625","51269531250","199951171875","479882812500","791806640625","950167968750","855151171875","586389375000","307854421875","123141768750","36942530625","8060188500","1209028275","111602610","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","64","512","4096","32768","262144","2097152","16777216","134217728","1073741824","8589934592","68719476736","549755813888","4398046511104"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161809,"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-66f5a589f0412003","e-69d63c555e6a484b"]},{"id":161810,"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-66f5a589f0412003","e-69d63c555e6a484b","e-ad86b3979b25a235"]},{"id":161811,"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-66f5a589f0412003","e-69d63c555e6a484b","e-ad86b3979b25a235"]},{"id":161812,"position":3,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-ad86b3979b25a235"]},{"id":161813,"position":4,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-ad86b3979b25a235"]}],"region_exact":{},"incoming_edges":[{"id":"e-66f5a589f0412003","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-05c39cd3844b146d","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":"8172042bbcc2e8eadffbc84a9beedecd5438e0f67b136f067e3a4d493e05b443","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-5733d79f45a14084","role":"sequence","name":"Triangle of coefficients in expansion of (2+3x)^n.","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]"}]},{"id":"e-69d63c555e6a484b","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-05c39cd3844b146d","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":"b19a0104a38446fda5fd5faed2f4166695548012d17ee7a7d98ad1048ecd04fe","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-3dcaebc50e901d48","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","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]"}]},{"id":"e-ad86b3979b25a235","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-05c39cd3844b146d","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":"86cb92a144f77eada0b91e3a0a41d46e11f61878b59c0862651cc31290b6eef7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5733d79f45a14084","role":"sequence","name":"Triangle of coefficients in expansion of (2+3x)^n.","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-3b40132009095e8e","kind":"fingerprint_related","node_a_id":"n-c44f7114310405ad","node_b_id":"n-05c39cd3844b146d","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-c44f7114310405ad","other":{"id":"n-c44f7114310405ad","name":"Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","5"],["9","30","25"]],"card_rows":[["1"],["3","5"],["9","30","25"],["27","135","225","125"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013622"],"derivation":{"root":"n-c44f7114310405ad","steps":[],"edges":[],"text":"A013622"},"headline":"A013622: Triangle of coefficients in expansion of (3+5x)^n.","chain":null}},{"id":"l-82e364a26afac62e","kind":"fingerprint_related","node_a_id":"n-05c39cd3844b146d","node_b_id":"n-384c17a53af3442c","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-384c17a53af3442c","other":{"id":"n-384c17a53af3442c","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","20"],["144","480","400"]],"card_rows":[["1"],["12","20"],["144","480","400"],["1728","8640","14400","8000"]],"row_sums_preview":["1","32","1024","32768","1048576","33554432","1073741824","34359738368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013611"],"edges":["e-62a1b0a86f60f435"],"text":"A038271 \u2192 basis A013611"},"headline":"A038271 \u2192 basis A013611","chain":null}},{"id":"l-b391a75342293292","kind":"fingerprint_related","node_a_id":"n-05c39cd3844b146d","node_b_id":"n-3ccd0708169ccfae","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-3ccd0708169ccfae","other":{"id":"n-3ccd0708169ccfae","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","270","81"],["3375","6075","3645","729"],["50625","121500","109350","43740","6561"]],"card_rows":[["225","270","81"],["3375","6075","3645","729"],["50625","121500","109350","43740","6561"],["759375","2278125","2733750","1640250","492075","59049"]],"row_sums_preview":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013610"],"edges":["e-a617f0e181e2ec6f"],"text":"A038329 \u2192 basis A013610"},"headline":"A038329 \u2192 basis A013610","chain":null}},{"id":"l-d5602f3148a068b4","kind":"fingerprint_related","node_a_id":"n-05c39cd3844b146d","node_b_id":"n-d524cbe046c3619e","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-d524cbe046c3619e","other":{"id":"n-d524cbe046c3619e","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","120","100"],["216","1080","1800","1000"],["1296","8640","21600","24000","10000"]],"card_rows":[["36","120","100"],["216","1080","1800","1000"],["1296","8640","21600","24000","10000"],["7776","64800","216000","360000","300000","100000"]],"row_sums_preview":["256","4096","65536","1048576","16777216","268435456","4294967296","68719476736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013609"],"edges":["e-31086658f6c15e80"],"text":"A013612 \u2192 basis A013609"},"headline":"A013612 \u2192 basis A013609","chain":null}},{"id":"l-8ac39fb46adf8a1c","kind":"oeis_match","node_a_id":"n-05c39cd3844b146d","node_b_id":null,"fingerprint":null,"oeis_id":"A013622","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"reversed_triangle","terms_compared":40},"in_catalog":true,"oeis_id":"A013622","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a5503bcfc15e73b6","kind":"oeis_match","node_a_id":"n-05c39cd3844b146d","node_b_id":null,"fingerprint":null,"oeis_id":"A038245","in_catalog":0,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":39},"in_catalog":false,"oeis_id":"A038245","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d0cef3a1bbc24a68","kind":"oeis_match","node_a_id":"n-05c39cd3844b146d","node_b_id":null,"fingerprint":null,"oeis_id":"A001018","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":"A001018","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A013622","relation":"match_reversed_triangle"},{"oeis_id":"A001018","relation":"match_row_sums"},{"oeis_id":"A038245","relation":"match_triangle"}],"bfiles":[{"node_id":"n-05c39cd3844b146d","oeis_id":"A001018","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":18961},{"node_id":"n-05c39cd3844b146d","oeis_id":"A013622","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":40,"oeis_terms_agreeing":40,"oeis_terms_available":40,"term_count":990,"raw_size":27318},{"node_id":"n-05c39cd3844b146d","oeis_id":"A038245","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":39,"oeis_terms_agreeing":39,"oeis_terms_available":39,"term_count":990,"raw_size":27318}],"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:       5,        3;\n  2:      25,       30,        9;\n  3:     125,      225,      135,       27;\n  4:     625,     1500,     1350,      540,       81;\n  5:    3125,     9375,    11250,     6750,     2025,      243;\n  6:   15625,    56250,    84375,    67500,    30375,     7290,     729;\n  7:   78125,   328125,   590625,   590625,   354375,   127575,   25515,    2187;\n  8:  390625,  1875000,  3937500,  4725000,  3543750,  1701000,  510300,   87480,   6561;\n  9: 1953125, 10546875, 25312500, 35437500, 31893750, 19136250, 7654500, 1968300, 295245, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5 + 3*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A013622","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     3,      5;\n  2:     9,     30,      25;\n  3:    27,    135,     225,     125;\n  4:    81,    540,    1350,    1500,      625;\n  5:   243,   2025,    6750,   11250,     9375,     3125;\n  6:   729,   7290,   30375,   67500,    84375,    56250,    15625;\n  7:  2187,  25515,  127575,  354375,   590625,   590625,   328125,    78125;\n  8:  6561,  87480,  510300, 1701000,  3543750,  4725000,  3937500,  1875000,   390625;\n  9: 19683, 295245, 1968300, 7654500, 19136250, 31893750, 35437500, 25312500, 10546875, 1953125;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (5+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5 + 3*x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (5 + 3*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A038245","target":"triangle","first_row":0,"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:       5,        3;\n  2:      25,       30,        9;\n  3:     125,      225,      135,       27;\n  4:     625,     1500,     1350,      540,       81;\n  5:    3125,     9375,    11250,     6750,     2025,      243;\n  6:   15625,    56250,    84375,    67500,    30375,     7290,     729;\n  7:   78125,   328125,   590625,   590625,   354375,   127575,   25515,    2187;\n  8:  390625,  1875000,  3937500,  4725000,  3543750,  1701000,  510300,   87480,   6561;\n  9: 1953125, 10546875, 25312500, 35437500, 31893750, 19136250, 7654500, 1968300, 295245, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5 + 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; for(n = 1, N, my(i = n + 1); v[i] = (5 + 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":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-5733d79f45a14084","steps":["basis A007318"],"edges":["e-66f5a589f0412003"],"text":"A013620 \u2192 basis A007318"},"headline":"A013620 \u2192 basis A007318","chain":null}
