{"id":"n-994b7fb19b3cbd3e","revision":0,"first_index":0,"fingerprint":"f6-df7467a067c1cbae","name":"theta_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) = x + 5.","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"],["5","10"],["25","120","75"],["125","1070","1605","500"],["625","8420","22470","16840","3125"],["3125","61770","258330","344440","154425","18750"],["15625","433120","2642205","5511040","4403675","1299360","109375"],["78125","2942470","24992235","75794780","94743475","49984470","10298645","625000"],["390625","19529820","223496400","939588040","1705382550","1409382060","521491600","78119280","3515625"],["1953125","127335170","1916011560","10794651520","27093293450","32511952140","18890640160","5109364160","573008265","19531250"],["9765625","818698520","15895116735","117013397120","392266099450","650239042800","549172539230","234026794240","47685350205","4093492600","107421875"],["48828125","5205159870","128458486065","1211622533700","5285595851850","11680302940740","13627020097530","8456953362960","2726150700825","428194953550","28628379285","585937500"],["244140625","32793459220","1016325499530","12090872009480","67259228839275","192915609674040","299794442145660","257220812898720","121066611910695","30227180023700","3726526831610","196760755320","3173828125"],["1220703125","205061536570","7901321517990","117053486066440","816923319719375","2978400066633090","5998557399038820","6855494170330080","4467600099949635","1633846639438750","321897086682710","31605286071960","1332899987705","17089843750"],["6103515625","1274314531920","60538319808465","1104989414248640","9547062741220125","43508312437616400","111156532400028885","164531860087921920","142915541657179995","72513854062694000","21003538030684275","3314968242745920","262332719170015","8920201723440","91552734375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","15","220","3300","51480","840840","14414400","259459200","4900896000","97037740800","2011327718400","43578767232000","985550582016000","23230835147520000","569929822285824000"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","10"]],"recurrence":{"denominator":{"coeffs":[["4"],["1"]]},"terms":[{"coeff":{"coeffs":[["-100","-120","-60"],["-20","-64","-47"],["0","-8","-12"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-20","0","20"],["0","-4","-5","9"],["0","0","-1","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["40","32"],["9","16"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"],["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188047,"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-b02b2a42240d5379"]},{"id":188048,"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-b02b2a42240d5379"]},{"id":188049,"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-b02b2a42240d5379"]},{"id":188050,"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-b02b2a42240d5379"]}],"region_exact":{},"incoming_edges":[{"id":"e-b02b2a42240d5379","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-994b7fb19b3cbd3e","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"18ada52e60952913a733ff1ab5f31fe077f91c4573cf623afc030dddf9850ade","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8919a5e7419ba068","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) = x + 5.","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:       5,        10;\n  2:      25,       120,         75;\n  3:     125,      1070,       1605,         500;\n  4:     625,      8420,      22470,       16840,        3125;\n  5:    3125,     61770,     258330,      344440,      154425,       18750;\n  6:   15625,    433120,    2642205,     5511040,     4403675,     1299360,      109375;\n  7:   78125,   2942470,   24992235,    75794780,    94743475,    49984470,    10298645,     625000;\n  8:  390625,  19529820,  223496400,   939588040,  1705382550,  1409382060,   521491600,   78119280,   3515625;\n  9: 1953125, 127335170, 1916011560, 10794651520, 27093293450, 32511952140, 18890640160, 5109364160, 573008265, 19531250;\n","recurrence_sentence":"The row polynomials satisfy (n+4)*P(n,x) = -(20*n+100+(8*n^2+64*n+120)*x+(n^3+12*n^2+47*n+60)*x^2)*P(n-2,x) - ((4*n+20)*x+(n^2+5*n)*x^2-(n^2+9*n+20)*x^3)*d/dx P(n-2,x) + (9*n+40+(2*n^2+16*n+32)*x)*P(n-1,x) + ((n+4)*x-(n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 10*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 10*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(20*n + 100 + (8*n^2 + 64*n + 120)*x + (n^3 + 12*n^2 + 47*n + 60)*x^2)*P[n - 2, x] - ((4*n + 20)*x + (n^2 + 5*n)*x^2 - (n^2 + 9*n + 20)*x^3)*D[P[n - 2, x], x] + (9*n + 40 + (2*n^2 + 16*n + 32)*x)*P[n - 1, x] + ((n + 4)*x - (n + 4)*x^2)*D[P[n - 1, x], x])/(n + 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 10*x; for(n = 2, N, my(i = n + 1); v[i] = (-(20*n + 100 + (8*n^2 + 64*n + 120)*x + (n^3 + 12*n^2 + 47*n + 60)*x^2)*v[i - 2] - ((4*n + 20)*x + (n^2 + 5*n)*x^2 - (n^2 + 9*n + 20)*x^3)*deriv(v[i - 2], x) + (9*n + 40 + (2*n^2 + 16*n + 32)*x)*v[i - 1] + ((n + 4)*x - (n + 4)*x^2)*deriv(v[i - 1], x))/(n + 4)); 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-8919a5e7419ba068","steps":["\u03b8+1"],"edges":["e-b02b2a42240d5379"],"text":"A257607 \u2192 \u03b8+1"},"headline":"A257607 \u2192 \u03b8+1","chain":null}