{"id":"n-56eee146d6109546","revision":0,"first_index":0,"fingerprint":"f6-e7b08fefd490fef1","name":"Triangle T(n,k) read by rows: T(n,k) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 1.","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"],["1","1"],["1","10","1"],["1","83","41","1"],["1","668","1110","122","1"],["1","5349","25982","8210","309","1"],["1","42798","572367","432328","44715","714","1"],["1","342391","12276495","20154955","4635787","202689","1561","1"],["1","2739136","260203132","879857170","402100930","38001292","815680","3298","1"],["1","21913097","5486178860","37015708724","31415703470","5658628682","260490608","3027488","6821","1"],["1","175304786","115406973933","1524517779840","2301130686618","731722574292","62938394274","1574800320","10617189","13922","1"],["1","1402438299","2425299500453","62019373958997","161769636868890","86353281938358","12761141884194","591856827690","8688068805","35749727","28185","1"],["1","11219506404","50946716330802","2505027953364410","11073200762108823","9584072161033488","2301704844166308","180733652978604","4914099619575","44762198780","116860266","56778","1"],["1","89756051245","1070015677023690","100761532014215222","744808329070717595","1019729734318194255","382799321562386868","47921056833398460","2190074332275999","37103913756275","218898145826","373746222","114037","1"],["1","718048409974","22471496046163675","4043301468692893160","49520918241753011117","105342137785253824430","60089877376661201739","11496500575177223904","828214122595888995","23529541845412994","260163836508041","1028081041880","1176090463","228634","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","12","126","1902","39852","1092924","37613880","1583720640","79861657752","4737477145176","325930717263600","25700656288224240","2299317476750025696","231367212519167899008"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A166972","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","7","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","3"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169589,"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-b97e7207b768496c"]},{"id":169590,"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-b97e7207b768496c"]},{"id":169591,"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-b97e7207b768496c"]}],"region_exact":{},"incoming_edges":[{"id":"e-b97e7207b768496c","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-56eee146d6109546","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; S_n = kappa (theta + a)(theta + b) + x (u + v theta) with kappa (t + a)(t + b) = 1 + (4) t + (3) t^2 and u + v theta = n + (-1) theta: The degree-local Euler step: T = kappa (theta + a)(theta + b) + x (u + v theta), theta = x D, with kappa, a, b > 0, u + v k > 0 for every k <= d and a negative comparison defect kappa (a b - (a + b + 1)^2/4) + (u - (a + b + 1) v/2) s - v^2 s^2/(4 kappa) < 0 for every s <= 0, maps every polynomial of degree <= d with nonnegative coefficients and only real roots to such a polynomial (RealRooted IsPFPolynomial.generalStep_of_natDegree_le, a half-line theorem: the symbol of T has discriminant -4 kappa s^2 times the comparison defect at s, so its real-rootedness is required for s <= 0 only, and the hypothesis (i) of P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5, for every real s, is not needed); here d = deg P_(n-1), so u(n) + v(n) k > 0 is required for 0 <= k <= deg P_(n-1) only (both ends, the condition being linear in k); by linearity a row with nonpositive coefficients is handled through its negative","lean_route":null,"lean_declarations":["RealRooted.IsPFPolynomial.generalStep_of_natDegree_le","RealRooted.EulerBidiagonal.generalStep","RealRooted.EulerBidiagonal.comparisonDefect","RealRooted.EulerBidiagonal.generalRowsDep_spec"],"certificate_hash":"82b5aded91e8948126ad810e3a0608b27fb7c6a2662fd8dc7927c54096e1f607","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-55fc88f76248181d","kind":"oeis_match","node_a_id":"n-56eee146d6109546","node_b_id":null,"fingerprint":null,"oeis_id":"A166972","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":43},"in_catalog":true,"oeis_id":"A166972","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A166972","relation":"identification"},{"oeis_id":"A166972","relation":"match_triangle"}],"bfiles":[{"node_id":"n-56eee146d6109546","oeis_id":"A166972","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":43,"oeis_terms_agreeing":43,"oeis_terms_available":43,"term_count":990,"raw_size":44656}],"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: 1,        1;\n  2: 1,       10,          1;\n  3: 1,       83,         41,           1;\n  4: 1,      668,       1110,         122,           1;\n  5: 1,     5349,      25982,        8210,         309,          1;\n  6: 1,    42798,     572367,      432328,       44715,        714,         1;\n  7: 1,   342391,   12276495,    20154955,     4635787,     202689,      1561,       1;\n  8: 1,  2739136,  260203132,   879857170,   402100930,   38001292,    815680,    3298,    1;\n  9: 1, 21913097, 5486178860, 37015708724, 31415703470, 5658628682, 260490608, 3027488, 6821, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+n*x)*P(n-1,x) + (7*x-x^2)*d/dx P(n-1,x) + 3*x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 1, x] + (7*x - x^2)*D[P[n - 1, x], x] + 3*x^2*D[P[n - 1, x], {x, 2}]];\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] = (1 + n*x)*v[i - 1] + (7*x - x^2)*deriv(v[i - 1], x) + 3*x^2*deriv(deriv(v[i - 1], x), 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":"A166972","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,        1;\n   3: 1,       10,          1;\n   4: 1,       83,         41,           1;\n   5: 1,      668,       1110,         122,           1;\n   6: 1,     5349,      25982,        8210,         309,          1;\n   7: 1,    42798,     572367,      432328,       44715,        714,         1;\n   8: 1,   342391,   12276495,    20154955,     4635787,     202689,      1561,       1;\n   9: 1,  2739136,  260203132,   879857170,   402100930,   38001292,    815680,    3298,    1;\n  10: 1, 21913097, 5486178860, 37015708724, 31415703470, 5658628682, 260490608, 3027488, 6821, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (7*x-x^2)*d/dx P(n-1,x) + 3*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (7*x - x^2)*D[P[n - 1, x], x] + 3*x^2*D[P[n - 1, x], {x, 2}]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (7*x - x^2)*deriv(v[i - 1], x) + 3*x^2*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #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":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-56eee146d6109546","steps":[],"edges":[],"text":"A166972"},"headline":"A166972: Triangle T(n,k) read by rows: T(n,k) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 1.","chain":null}