{"id":"n-f03980d46a617c93","revision":1,"first_index":0,"fingerprint":"f6-70eb621354118c3d","name":"reversal of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= 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":false,"rows":[["2"],["4","2"],["8","12","2"],["16","48","28","2"],["32","160","200","60","2"],["64","480","1040","720","124","2"],["128","1344","4480","5600","2408","252","2"],["256","3584","17024","33600","27216","7728","508","2"],["512","9216","59136","169344","222432","124320","24200","1020","2"],["1024","23040","192000","752640","1460928","1360800","545680","74640","2044","2"],["2048","56320","591360","3041280","8190336","11487168","7895360","2332000","228008","4092","2"],["4096","135168","1745920","11404800","40710912","80306688","84713728","44140800","9784016","692208","8188","2"],["8192","319488","4978688","40268800","184065024","486300672","731574272","596563968","240272032","40520480","2093000","16380","2"],["16384","745472","13791232","135303168","770818048","2629186560","5353553920","6314147840","4059927872","1282401120","166267920","6311760","32764","2"],["32768","1720320","37273600","436101120","3029970944","12966553600","34369786880","55456727040","52318890624","26924369472","6744541440","677695200","19000808","65532","2"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","6","22","94","454","2430","14214","89918","610182","4412798","33827974","273646526","2326980998","20732504062","192982729350"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165912,"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-88af8bd6d056b310"]},{"id":165913,"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-88af8bd6d056b310"]},{"id":165914,"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-88af8bd6d056b310"]}],"region_exact":{},"incoming_edges":[{"id":"e-88af8bd6d056b310","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-f03980d46a617c93","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":"7242e9b8faa31073e258a6e5de11756b4f2f070381231a29849295ec9135d504","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-d5fb64ec7bd8ef70","role":"sequence","name":"Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= 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-91c1b203b75a5ee1","kind":"fingerprint_related","node_a_id":"n-d5fb64ec7bd8ef70","node_b_id":"n-f03980d46a617c93","fingerprint":"f6-70eb621354118c3d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-70eb621354118c3d"},"other_id":"n-d5fb64ec7bd8ef70","other":{"id":"n-d5fb64ec7bd8ef70","name":"Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","first_index":0,"fingerprint":"f6-70eb621354118c3d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2","4"],["2","12","8"]],"card_rows":[["2"],["2","4"],["2","12","8"],["2","28","48","16"]],"row_sums_preview":["2","6","22","94","454","2430","14214","89918"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A227450"],"derivation":{"root":"n-d5fb64ec7bd8ef70","steps":[],"edges":[],"text":"A227450"},"headline":"A227450: Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","chain":null}},{"id":"l-2a23e496c598976f","kind":"oeis_match","node_a_id":"n-f03980d46a617c93","node_b_id":null,"fingerprint":null,"oeis_id":"A227450","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A227450","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b3e02c577d816aa0","kind":"oeis_match","node_a_id":"n-f03980d46a617c93","node_b_id":null,"fingerprint":null,"oeis_id":"A001861","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001861","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A227450","relation":"match_reversed_triangle"},{"oeis_id":"A001861","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-f03980d46a617c93","oeis_id":"A227450","target":"reversed_triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":25788}],"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:    2;\n  1:    4,     2;\n  2:    8,    12,      2;\n  3:   16,    48,     28,      2;\n  4:   32,   160,    200,     60,       2;\n  5:   64,   480,   1040,    720,     124,       2;\n  6:  128,  1344,   4480,   5600,    2408,     252,      2;\n  7:  256,  3584,  17024,  33600,   27216,    7728,    508,     2;\n  8:  512,  9216,  59136, 169344,  222432,  124320,  24200,  1020,    2;\n  9: 1024, 23040, 192000, 752640, 1460928, 1360800, 545680, 74640, 2044, 2;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+n*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.","mathematica":"P[0, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[(2 + n*x)*P[n - 1, x] - 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] = 2; for(n = 1, N, my(i = n + 1); v[i] = (2 + n*x)*v[i - 1] - 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}},{"oeis_id":"A227450","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 2;\n   2: 2,    4;\n   3: 2,   12,     8;\n   4: 2,   28,    48,     16;\n   5: 2,   60,   200,    160,      32;\n   6: 2,  124,   720,   1040,     480,      64;\n   7: 2,  252,  2408,   5600,    4480,    1344,    128;\n   8: 2,  508,  7728,  27216,   33600,   17024,   3584,    256;\n   9: 2, 1020, 24200, 124320,  222432,  169344,  59136,   9216,   512;\n  10: 2, 2044, 74640, 545680, 1360800, 1460928, 752640, 192000, 23040, 1024;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+(n-1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.","mathematica":"P[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[(2 + (n - 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = (2 + (n - 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-d5fb64ec7bd8ef70","steps":["reverse"],"edges":["e-88af8bd6d056b310"],"text":"A227450 \u2192 reverse"},"headline":"A227450 \u2192 reverse","chain":null}