{"id":"n-73540840b1c566dc","revision":0,"first_index":0,"fingerprint":"f6-86218b05355409c1","name":"shift_x_plus_one of shift_x_plus_one of Triangle read by rows: A008277 * A007318.","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"],["4","1"],["19","9","1"],["103","70","15","1"],["622","543","169","22","1"],["4117","4351","1755","335","30","1"],["29521","36402","17896","4460","590","39","1"],["227290","318907","183918","56196","9765","959","49","1"],["1865881","2925333","1928179","694470","148701","19278","1470","60","1"],["16239523","28061620","20744637","8573881","2173248","348663","35238","2154","72","1"],["149142952","281015445","229694371","106840780","31189810","5945499","746277","60660","3045","85","1"],["1439618143","2932386403","2620748691","1351857551","445541745","98132287","14682129","1486617","99495","4180","99","1"],["14555631781","31826042304","30823596508","17430252488","6388175816","1593010650","276172534","33422796","2793417","156805","5599","114","1"],["153700654036","358627422349","373639893744","229493473716","92430818766","25696377278","5056614706","710865298","71177535","4999852","238953","7345","130","1"],["1690684883191","4188677128245","4665908048257","3089264034948","1354385683014","414717278976","91190695596","14584393824","1699991722","143344201","8587579","353808","9464","147","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","29","189","1357","10589","88909","797085","7583373","76179037","804638925","8904557341","102929260813","1239432543709","15511264432973"],"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":[["4","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180752,"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-2cf6d86d3c754dfa"]},{"id":180753,"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-2cf6d86d3c754dfa"]},{"id":180754,"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-2cf6d86d3c754dfa"]}],"region_exact":{},"incoming_edges":[{"id":"e-2cf6d86d3c754dfa","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-73540840b1c566dc","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":"9bd1ec42a4eab8530b4f2ce4257f52e42acfefcf9cd2efdccb0bba2493076e6f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-6186cc445ed1db3e","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: A008277 * A007318.","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":[],"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:        4,        1;\n  2:       19,        9,        1;\n  3:      103,       70,       15,       1;\n  4:      622,      543,      169,      22,       1;\n  5:     4117,     4351,     1755,     335,      30,      1;\n  6:    29521,    36402,    17896,    4460,     590,     39,     1;\n  7:   227290,   318907,   183918,   56196,    9765,    959,    49,    1;\n  8:  1865881,  2925333,  1928179,  694470,  148701,  19278,  1470,   60,  1;\n  9: 16239523, 28061620, 20744637, 8573881, 2173248, 348663, 35238, 2154, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + (3+x)*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[(4 + x)*P[n - 1, x] + (3 + x)*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] = (4 + x)*v[i - 1] + (3 + x)*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-bdae5aa3d1cac0ec","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-73e4f5467288c90f","e-2cf6d86d3c754dfa"],"text":"A137597 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A137597 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}
