{"id":"n-b706353c3a058005","revision":0,"first_index":0,"fingerprint":"f6-cef044b741d89456","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","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":false,"rows":[["1"],["12","6"],["180","180","36"],["3240","4860","2052","216"],["68040","136080","89100","21060","1296"],["1632960","4082400","3645000","1385100","205092","7776"],["44089920","132269760","150028200","79606800","19697580","1939140","46656"],["1322697600","4629441600","6378341760","4372250400","1548658440","265458060","18012132","279936"],["43649020800","174596083200","283321825920","238879186560","111222334440","28008121680","3455678700","165468420","1679616"],["1571364748800","7071141369600","13213749024000","13249461859200","7685798944320","2590172575200","482324041800","43950170700","1509371172","10077696"],["61283225203200","306416126016000","648187958880000","754255079424000","525503656117440","223565919635520","56751911231400","8025963699600","550042615980","13705272900","60466176"],["2573895458534400","14156425021939200","33446498678208000","44336056387392000","36129996827971200","18648367576977600","6056930181899520","1190079639856800","130290222581640","6806090485260","124073050212","362797056"],["115825295634048000","694951773804288000","1814044749239923200","2699832486326976000","2522371586944809600","1534960941664550400","611097936758749440","155757445176453120","24157528191636840","2076862967459280","83534181576300","1121011016580","2176782336"],["5559614190434304000","36137492237822976000","103292998646444006400","170599077939389299200","179899849178803353600","126324858683274451200","59776883527705344000","18871801501974220800","3850687025304799680","478449629766108000","32656559780262600","1019225344164300","10115220537252","13060694016"],["283540323712149504000","1984782265985046528000","6165959613078545280000","11193588220665666816000","13157660923791345561600","10484389935498390912000","5762382916912509888000","2180129900307270912000","556891785730610890560","92347218645787385280","9297881091175458600","508194452898896400","12382432438030380","91193713163460","78364164096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","396","10368","315576","10958328","427678056","18535139928","883299399336","45909482182488","2584553648562216","156675470521692888","10175161264078269096","704825398474324521048","51871491653575913390376"],"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":[["9","3"],["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191381,"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-fbf5d8d30514347e"]},{"id":191382,"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-fbf5d8d30514347e"]},{"id":191383,"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-fbf5d8d30514347e"]},{"id":191384,"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-fbf5d8d30514347e"]}],"region_exact":{},"incoming_edges":[{"id":"e-fbf5d8d30514347e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b706353c3a058005","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":"222ff68086dcfb1989798fdf32c221894ddc3ceb6c13fb1067dde8232620557e","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-61cf3a32da4a0e22","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","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-196a6c7b34e79c6f","kind":"fingerprint_related","node_a_id":"n-b706353c3a058005","node_b_id":"n-baf71172ebf4970d","fingerprint":"f6-cef044b741d89456","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cef044b741d89456"},"other_id":"n-baf71172ebf4970d","other":{"id":"n-baf71172ebf4970d","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","first_index":0,"fingerprint":"f6-cef044b741d89456","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","12"],["36","180","180"]],"card_rows":[["1"],["6","12"],["36","180","180"],["216","2052","4860","3240"]],"row_sums_preview":["1","18","396","10368","315576","10958328","427678056","18535139928"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-61cf3a32da4a0e22","steps":["x \u21a6 x/(x+1)"],"edges":["e-4e8b651ed87c3dde"],"text":"A257626 \u2192 x \u21a6 x/(x+1)"},"headline":"A257626 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-5ae49e31a5a6897b","kind":"fingerprint_related","node_a_id":"n-a62fc9bae1fba01f","node_b_id":"n-b706353c3a058005","fingerprint":"f6-cef044b741d89456","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cef044b741d89456"},"other_id":"n-a62fc9bae1fba01f","other":{"id":"n-a62fc9bae1fba01f","name":"shift_x_plus_one of Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.","first_index":1,"fingerprint":"f6-cef044b741d89456","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["20","20","4"],["120","180","76","8"],["840","1680","1100","260","16"]],"card_rows":[["20","20","4"],["120","180","76","8"],["840","1680","1100","260","16"],["6720","16800","15000","5700","844","32"]],"row_sums_preview":["44","384","3896","45096","586664","8475144","134628776","2332443336"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c8a3637306e75dc","steps":["x \u21a6 x+1"],"edges":["e-92dd70260befe006"],"text":"A256890 \u2192 x \u21a6 x+1"},"headline":"A256890 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-89ca7c24a62dd865","kind":"fingerprint_related","node_a_id":"n-6497b30746d46dc5","node_b_id":"n-b706353c3a058005","fingerprint":"f6-cef044b741d89456","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cef044b741d89456"},"other_id":"n-6497b30746d46dc5","other":{"id":"n-6497b30746d46dc5","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.","first_index":0,"fingerprint":"f6-cef044b741d89456","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","4"],["80","80","16"]],"card_rows":[["1"],["8","4"],["80","80","16"],["960","1440","608","64"]],"row_sums_preview":["1","12","176","3072","62336","1443072","37546496","1084818432"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c35e92273dac3e06","steps":["x \u21a6 x+1"],"edges":["e-176cfd7833813147"],"text":"A257613 \u2192 x \u21a6 x+1"},"headline":"A257613 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-c55b0af56bdf396e","kind":"fingerprint_related","node_a_id":"n-96cfdd1a383cbf1a","node_b_id":"n-b706353c3a058005","fingerprint":"f6-cef044b741d89456","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cef044b741d89456"},"other_id":"n-96cfdd1a383cbf1a","other":{"id":"n-96cfdd1a383cbf1a","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.","first_index":0,"fingerprint":"f6-cef044b741d89456","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","8"],["16","80","80"]],"card_rows":[["1"],["4","8"],["16","80","80"],["64","608","1440","960"]],"row_sums_preview":["1","12","176","3072","62336","1443072","37546496","1084818432"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c35e92273dac3e06","steps":["x \u21a6 x/(x+1)"],"edges":["e-33febc4c05c90aa9"],"text":"A257613 \u2192 x \u21a6 x/(x+1)"},"headline":"A257613 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f461c3c9efc1de90","kind":"fingerprint_related","node_a_id":"n-0b37b712dda89f30","node_b_id":"n-b706353c3a058005","fingerprint":"f6-cef044b741d89456","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cef044b741d89456"},"other_id":"n-0b37b712dda89f30","other":{"id":"n-0b37b712dda89f30","name":"mobius_x_over_x_plus_one of Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.","first_index":1,"fingerprint":"f6-cef044b741d89456","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["4","20","20"],["8","76","180","120"],["16","260","1100","1680","840"]],"card_rows":[["4","20","20"],["8","76","180","120"],["16","260","1100","1680","840"],["32","844","5700","15000","16800","6720"]],"row_sums_preview":["44","384","3896","45096","586664","8475144","134628776","2332443336"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c8a3637306e75dc","steps":["x \u21a6 x/(x+1)"],"edges":["e-cd779c54ec89ccb2"],"text":"A256890 \u2192 x \u21a6 x/(x+1)"},"headline":"A256890 \u2192 x \u21a6 x/(x+1)","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:            12,             6;\n  2:           180,           180,             36;\n  3:          3240,          4860,           2052,            216;\n  4:         68040,        136080,          89100,          21060,          1296;\n  5:       1632960,       4082400,        3645000,        1385100,        205092,          7776;\n  6:      44089920,     132269760,      150028200,       79606800,      19697580,       1939140,        46656;\n  7:    1322697600,    4629441600,     6378341760,     4372250400,    1548658440,     265458060,     18012132,      279936;\n  8:   43649020800,  174596083200,   283321825920,   238879186560,  111222334440,   28008121680,   3455678700,   165468420,    1679616;\n  9: 1571364748800, 7071141369600, 13213749024000, 13249461859200, 7685798944320, 2590172575200, 482324041800, 43950170700, 1509371172, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+9+(3*n+3)*x)*P(n-1,x) - (3*x+3*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n + 9 + (3*n + 3)*x)*P[n - 1, x] - (3*x + 3*x^2)*D[P[n - 1, x], 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] = (3*n + 9 + (3*n + 3)*x)*v[i - 1] - (3*x + 3*x^2)*deriv(v[i - 1], x)); 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-61cf3a32da4a0e22","steps":["x \u21a6 x+1"],"edges":["e-fbf5d8d30514347e"],"text":"A257626 \u2192 x \u21a6 x+1"},"headline":"A257626 \u2192 x \u21a6 x+1","chain":null}
