{"id":"n-6a7b824517cedbda","revision":0,"first_index":0,"fingerprint":"f6-8ca1e631330e3f91","name":"reversal of basis_map_A013609 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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"],["2","16"],["4","64","256"],["8","192","1536","4096"],["16","512","6144","32768","65536"],["32","1280","20480","163840","655360","1048576"],["64","3072","61440","655360","3932160","12582912","16777216"],["128","7168","172032","2293760","18350080","88080384","234881024","268435456"],["256","16384","458752","7340032","73400320","469762048","1879048192","4294967296","4294967296"],["512","36864","1179648","22020096","264241152","2113929216","11274289152","38654705664","77309411328","68719476736"],["1024","81920","2949120","62914560","880803840","8455716864","56371445760","257698037760","773094113280","1374389534720","1099511627776"],["2048","180224","7208960","173015040","2768240640","31004295168","248034361344","1417339207680","5669356830720","15118284881920","24189255811072","17592186044416"],["4096","393216","17301504","461373440","8304721920","106300440576","992137445376","6803228196864","34016140984320","120946279055360","290271069732864","422212465065984","281474976710656"],["8192","851968","40894464","1199570944","23991418880","345476431872","3685081939968","29480655519744","176883933118464","786150813859840","2515682604351488","5488762045857792","7318349394477056","4503599627370496"],["16384","1835008","95420416","3053453312","67175972864","1074815565824","12897786789888","117922622078976","825458354552832","4402444557615104","17609778230460416","51228445761339392","102456891522678784","126100789566373888","72057594037927936"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","324","5832","104976","1889568","34012224","612220032","11019960576","198359290368","3570467226624","64268410079232","1156831381426176","20822964865671168","374813367582081024"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190351,"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-0dd7061f08e95988"]},{"id":190352,"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-0dd7061f08e95988"]},{"id":190353,"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-0dd7061f08e95988"]}],"region_exact":{},"incoming_edges":[{"id":"e-0dd7061f08e95988","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-6a7b824517cedbda","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"46423c0ce7f41a2a835fb2d85cb827047f6a4999dbf4fe8c04d57755b27185bd","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-326a4380cc213ab2","role":"sequence","name":"basis_map_A013609 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,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-21576fafa655eec4","kind":"fingerprint_related","node_a_id":"n-30f51be2f0e255b7","node_b_id":"n-6a7b824517cedbda","fingerprint":"f6-8ca1e631330e3f91","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8ca1e631330e3f91"},"other_id":"n-30f51be2f0e255b7","other":{"id":"n-30f51be2f0e255b7","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-8ca1e631330e3f91","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["16","2"]],"card_rows":[["1"],["16","2"],["192","48","3"],["2048","768","96","4"]],"row_sums_preview":["0","1","18","243","2916","32805","354294","3720087"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["D"],"edges":["e-cd7cfb2c92703982"],"text":"A038279 \u2192 D"},"headline":"A038279 \u2192 D","chain":null}},{"id":"l-2548700a8c4aa7e3","kind":"fingerprint_related","node_a_id":"n-4c9e2de513dce1e4","node_b_id":"n-6a7b824517cedbda","fingerprint":"f6-8ca1e631330e3f91","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8ca1e631330e3f91"},"other_id":"n-4c9e2de513dce1e4","other":{"id":"n-4c9e2de513dce1e4","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-8ca1e631330e3f91","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","8"],["1","16","64"]],"card_rows":[["1"],["1","8"],["1","16","64"],["1","24","192","512"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["reverse"],"edges":["e-7d675385f679e5a5"],"text":"A038279 \u2192 reverse"},"headline":"A038279 \u2192 reverse","chain":null}},{"id":"l-7dfe7418c200746c","kind":"fingerprint_related","node_a_id":"n-326a4380cc213ab2","node_b_id":"n-6a7b824517cedbda","fingerprint":"f6-8ca1e631330e3f91","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8ca1e631330e3f91"},"other_id":"n-326a4380cc213ab2","other":{"id":"n-326a4380cc213ab2","name":"basis_map_A013609 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-8ca1e631330e3f91","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","2"],["256","64","4"]],"card_rows":[["1"],["16","2"],["256","64","4"],["4096","1536","192","8"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013609"],"edges":["e-f16669f3777d4fa6"],"text":"A027467 \u2192 basis A013609"},"headline":"A027467 \u2192 basis A013609","chain":null}},{"id":"l-83bc20aa67b7bb96","kind":"fingerprint_related","node_a_id":"n-6a7b824517cedbda","node_b_id":"n-d89fc0d0973b1556","fingerprint":"f6-8ca1e631330e3f91","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8ca1e631330e3f91"},"other_id":"n-d89fc0d0973b1556","other":{"id":"n-d89fc0d0973b1556","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-8ca1e631330e3f91","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","64","4"],["4096","1536","192","8"],["65536","32768","6144","512","16"]],"card_rows":[["256","64","4"],["4096","1536","192","8"],["65536","32768","6144","512","16"],["1048576","655360","163840","20480","1280","32"]],"row_sums_preview":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-02ed32e9293d9244","e-56a43e4060946796"],"text":"A038328 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A038328 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-864f74b1a35b3f20","kind":"fingerprint_related","node_a_id":"n-798901ae44ad10ac","node_b_id":"n-6a7b824517cedbda","fingerprint":"f6-8ca1e631330e3f91","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8ca1e631330e3f91"},"other_id":"n-798901ae44ad10ac","other":{"id":"n-798901ae44ad10ac","name":"derivative of basis_map_A013609 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-8ca1e631330e3f91","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["64","8"]],"card_rows":[["2"],["64","8"],["1536","384","24"],["32768","12288","1536","64"]],"row_sums_preview":["0","2","72","1944","46656","1049760","22674816","476171136"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013609","D"],"edges":["e-f16669f3777d4fa6","e-730533ba72f0a4fa"],"text":"A027467 \u2192 basis A013609 \u2192 D"},"headline":"A027467 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-fd0a77a5ae9c01b0","kind":"fingerprint_related","node_a_id":"n-366b2525683554ec","node_b_id":"n-6a7b824517cedbda","fingerprint":"f6-8ca1e631330e3f91","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8ca1e631330e3f91"},"other_id":"n-366b2525683554ec","other":{"id":"n-366b2525683554ec","name":"Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-8ca1e631330e3f91","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","1"],["64","16","1"]],"card_rows":[["1"],["8","1"],["64","16","1"],["512","192","24","1"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038279"],"derivation":{"root":"n-366b2525683554ec","steps":[],"edges":[],"text":"A038279"},"headline":"A038279: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","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:   2,    16;\n  2:   4,    64,     256;\n  3:   8,   192,    1536,     4096;\n  4:  16,   512,    6144,    32768,     65536;\n  5:  32,  1280,   20480,   163840,    655360,    1048576;\n  6:  64,  3072,   61440,   655360,   3932160,   12582912,    16777216;\n  7: 128,  7168,  172032,  2293760,  18350080,   88080384,   234881024,   268435456;\n  8: 256, 16384,  458752,  7340032,  73400320,  469762048,  1879048192,  4294967296,  4294967296;\n  9: 512, 36864, 1179648, 22020096, 264241152, 2113929216, 11274289152, 38654705664, 77309411328, 68719476736;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+16*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[(2 + 16*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] = (2 + 16*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-ff0214c333bfdcca","steps":["basis A013609","reverse"],"edges":["e-f16669f3777d4fa6","e-0dd7061f08e95988"],"text":"A027467 \u2192 basis A013609 \u2192 reverse"},"headline":"A027467 \u2192 basis A013609 \u2192 reverse","chain":null}
