{"id":"n-6d0748781e5e30ed","revision":0,"first_index":0,"fingerprint":"f6-9111d60e4a92b32c","name":"shift_x_plus_one of basis_map_A013610 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"],["14","9"],["196","252","81"],["2744","5292","3402","729"],["38416","98784","95256","40824","6561"],["537824","1728720","2222640","1428840","459270","59049"],["7529536","29042496","46675440","40007520","19289340","4960116","531441"],["105413504","474360768","914838624","980184240","630118440","243045684","52081218","4782969"],["1475789056","7589772288","17076987648","21956126976","17643316320","9073705536","2916548208","535692528","43046721"],["20661046784","119538913536","307385777664","461078666496","444611571264","285821724384","122495024736","33748629264","5423886846","387420489"],["289254654976","1859494210560","5379251109120","9221573329920","10374269996160","8003008282752","4287325865760","1574936032320","379672079220","54238868460","3486784401"],["4049565169664","28636210842624","92044963422720","177515286600960","228233939915520","205410545923968","132049636665408","60635037244320","19489833399960","4176392871420","536964797754","31381059609"],["56693912375296","437353038323712","1546355385501696","3313618683217920","4792912738225920","4929853102175232","3697389826631424","2037337251409152","818573002798320","233878000799520","45105043011336","5272018014312","282429536481"],["793714773254144","6633187747909632","25585152741937152","60307860034566144","96923346484124160","112154158074486528","96132135492417024","61799229959410944","29796057301858848","10641449036378160","2736372609354384","479753639302392","51402175639542","2541865828329"],["11112006825558016","100008061430022144","417890828118306816","1074576415161360384","1899697591088833536","2442468331399928832","2355237319564217088","1730378438863506432","973337871860722368","417144802226023872","134082257858364816","31343904434422944","5037413212675116","498205702352484","22876792454961"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","23","529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463","41426511213649","952809757913927","21914624432020321","504036361936467383","11592836324538749809"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["14","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188155,"position":0,"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-80181af021c9a6e2"]},{"id":188156,"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-08a03f724769d7c5","e-80181af021c9a6e2","e-f9f820b41f38b70d"]},{"id":188157,"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-08a03f724769d7c5","e-80181af021c9a6e2","e-f9f820b41f38b70d"]},{"id":188158,"position":3,"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-80181af021c9a6e2"]},{"id":188159,"position":4,"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-08a03f724769d7c5","e-f9f820b41f38b70d"]}],"region_exact":{},"incoming_edges":[{"id":"e-08a03f724769d7c5","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-6d0748781e5e30ed","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":"501a82224daba7d63e3b4f740cb7dbfe17dab3317b26733100489087646a282a","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-3c11bfaa4e68c9bc","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^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-f9f820b41f38b70d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6d0748781e5e30ed","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":"122b3a7dbaae8b440f0936625722ab42a79c8b254e1622e48fe26f99e9632b7e","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-a13c2de84911142b","role":"sequence","name":"basis_map_A013610 of 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-80181af021c9a6e2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-6d0748781e5e30ed","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":"c414eabe86d665869d2652ee6b0e3080070d33e1f29461329d3ab09c1c5ed8b5","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-a13c2de84911142b","role":"sequence","name":"basis_map_A013610 of 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-201128e6f38f86bf","kind":"fingerprint_related","node_a_id":"n-6d0748781e5e30ed","node_b_id":"n-1bb976829c17ec98","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-1bb976829c17ec98","other":{"id":"n-1bb976829c17ec98","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1008","1568"],["27216","84672","65856"],["653184","3048192","4741632","2458624"]],"card_rows":[["1008","1568"],["27216","84672","65856"],["653184","3048192","4741632","2458624"],["14696640","91445760","213373440","221276160","86051840"]],"row_sums_preview":["2576","177744","10901632","626843840","34601779968","1856962191616","97623155216384","5051998282447872"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013611","D"],"edges":["e-e460f89b751823ed","e-27550b5f4421b741"],"text":"A038321 \u2192 basis A013611 \u2192 D"},"headline":"A038321 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-4192ca0ce32d0170","kind":"fingerprint_related","node_a_id":"n-4b2df01c2184142e","node_b_id":"n-6d0748781e5e30ed","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-4b2df01c2184142e","other":{"id":"n-4b2df01c2184142e","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","1008","784"],["5832","27216","42336","21952"],["104976","653184","1524096","1580544","614656"]],"card_rows":[["324","1008","784"],["5832","27216","42336","21952"],["104976","653184","1524096","1580544","614656"],["1889568","14696640","45722880","71124480","55319040","17210368"]],"row_sums_preview":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013611"],"edges":["e-e460f89b751823ed"],"text":"A038321 \u2192 basis A013611"},"headline":"A038321 \u2192 basis A013611","chain":null}},{"id":"l-64fa174d6fa2aed6","kind":"fingerprint_related","node_a_id":"n-6d0748781e5e30ed","node_b_id":"n-ce03950bfb372f88","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-ce03950bfb372f88","other":{"id":"n-ce03950bfb372f88","name":"basis_map_A013610 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["196","252","81"],["2744","5292","3402","729"],["38416","98784","95256","40824","6561"]],"card_rows":[["196","252","81"],["2744","5292","3402","729"],["38416","98784","95256","40824","6561"],["537824","1728720","2222640","1428840","459270","59049"]],"row_sums_preview":["529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-108b34f10b7468f4","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-3a9169eda66e0655","e-066880bf2751c810"],"text":"A038281 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"A038281 \u2192 x \u21a6 x+1 \u2192 basis A013610","chain":null}},{"id":"l-7957ab7e5b0f6cfe","kind":"fingerprint_related","node_a_id":"n-6d0748781e5e30ed","node_b_id":"n-8c08995d48487910","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-8c08995d48487910","other":{"id":"n-8c08995d48487910","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["784","1008","324"],["21952","42336","27216","5832"],["614656","1580544","1524096","653184","104976"]],"card_rows":[["784","1008","324"],["21952","42336","27216","5832"],["614656","1580544","1524096","653184","104976"],["17210368","55319040","71124480","45722880","14696640","1889568"]],"row_sums_preview":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013611","reverse"],"edges":["e-e460f89b751823ed","e-541bf93e05058bc8"],"text":"A038321 \u2192 basis A013611 \u2192 reverse"},"headline":"A038321 \u2192 basis A013611 \u2192 reverse","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:          14,            9;\n  2:         196,          252,           81;\n  3:        2744,         5292,         3402,          729;\n  4:       38416,        98784,        95256,        40824,         6561;\n  5:      537824,      1728720,      2222640,      1428840,       459270,        59049;\n  6:     7529536,     29042496,     46675440,     40007520,     19289340,      4960116,       531441;\n  7:   105413504,    474360768,    914838624,    980184240,    630118440,    243045684,     52081218,     4782969;\n  8:  1475789056,   7589772288,  17076987648,  21956126976,  17643316320,   9073705536,   2916548208,   535692528,   43046721;\n  9: 20661046784, 119538913536, 307385777664, 461078666496, 444611571264, 285821724384, 122495024736, 33748629264, 5423886846, 387420489;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (14+9*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[(14 + 9*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] = (14 + 9*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}}],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013610","basis A013610"],"edges":["e-254f1b828fe6620e","e-08a03f724769d7c5"],"text":"A038303 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038303 \u2192 basis A013610 \u2192 basis A013610","chain":null}