{"id":"n-6f6f9db0a6c695a0","revision":0,"first_index":1,"fingerprint":"f6-4ace4b2db4280241","name":"theta_plus_one_squared of Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","8"],["1","64","108"],["1","264","2556","1920"],["1","896","25848","95232","42000"],["1","2824","189720","1861376","3651600","1088640"],["1","8640","1190052","24497152","120116400","148690944","32598720"],["1","26120","6844500","260337280","2535566000","7516499328","6495330240","1107025920"],["1","78592","37396080","2430760960","41531836000","236032524288","473209341696","305051074560","42032390400"],["1","236040","197943984","20871278080","578881980000","5586649959168","20929980046080","30534678577152","15385600753920","1764322560000"],["1","708416","1026896220","169324374016","7231842884000","109801372520448","682708512597888","1824876018401280","2039246031230208","831294019584000","81123551308800"],["1","2125576","5258808108","1320364520832","83606439364000","1895129445420288","18141226489248128","79091793312006144","159386052913526016","141673971015936000","47973419707530240","4054836680294400"],["1","6377088","26705339496","10009169005568","913377380252400","29755526078251008","416399666521103104","2750224668875685888","8915797055981424384","14104633929530572800","10264419080039393280","2947877953516339200","218904863559782400"],["1","19131656","134875451592","74337037358336","9565378507394800","435162387920964480","8572240683415322368","81499672495256600576","395221266709889511168","994460397370678425600","1273682438074210830336","776300599182851112960","192294386893092556800","12693891496366080000"],["1","57395392","678767973012","543857562407936","97006658928558800","6026459321688975360","162377303959369866304","2140599558991371698176","14764307328209234227968","55001381792674771353600","110978126856854890085376","117896879826307226271744","61294010215646160875520","13277661639636654489600","786891743269632000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","173","4741","163977","6794161","327101909","17921629389","1100325402577","74801183334425","5576451700505277","452301361628779733","39648936157363526617","3735169695144120740673","376308360276066316178789"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","8"]],"recurrence":{"denominator":{"coeffs":[["81"],["-72"],["16"]]},"terms":[{"coeff":{"coeffs":[["45","3256","174636"],["-56","-6112","-347040"],["16","3200","251648"],["0","-512","-79360"],["0","0","9216"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-156","-2248","124416"],["0","224","3808","-229824"],["0","-64","-2432","151296"],["0","0","512","-41984"],["0","0","0","4096"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-164","-6800","14256"],["0","0","224","11072","-23040"],["0","0","-64","-5888","12032"],["0","0","0","1024","-2048"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["36","-4536"],["-16","6624"],["0","-3200"],["0","512"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","244","3240"],["0","-256","-5472"],["0","64","2944"],["0","0","-512"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191181,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-3bb79e8069e7d5c6"]},{"id":191182,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-3bb79e8069e7d5c6"]},{"id":191183,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-3bb79e8069e7d5c6"]},{"id":191184,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-3bb79e8069e7d5c6"]}],"region_exact":{},"incoming_edges":[{"id":"e-3bb79e8069e7d5c6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-6f6f9db0a6c695a0","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":"55ad9891aba8526c63e7f704a1d9f85703b727a9fced46055ce9a19af8757560","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-945ed98594b7abe2","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,      8;\n   3: 1,     64,       108;\n   4: 1,    264,      2556,        1920;\n   5: 1,    896,     25848,       95232,        42000;\n   6: 1,   2824,    189720,     1861376,      3651600,       1088640;\n   7: 1,   8640,   1190052,    24497152,    120116400,     148690944,       32598720;\n   8: 1,  26120,   6844500,   260337280,   2535566000,    7516499328,     6495330240,     1107025920;\n   9: 1,  78592,  37396080,  2430760960,  41531836000,  236032524288,   473209341696,   305051074560,    42032390400;\n  10: 1, 236040, 197943984, 20871278080, 578881980000, 5586649959168, 20929980046080, 30534678577152, 15385600753920, 1764322560000;\n","recurrence_sentence":"The row polynomials satisfy (16*n^2-72*n+81)*P(n,x) = (16*n^2-56*n+45-(512*n^3-3200*n^2+6112*n-3256)*x+(9216*n^4-79360*n^3+251648*n^2-347040*n+174636)*x^2)*P(n-2,x) - ((64*n^2-224*n+156)*x-(512*n^3-2432*n^2+3808*n-2248)*x^2-(4096*n^4-41984*n^3+151296*n^2-229824*n+124416)*x^3)*d/dx P(n-2,x) - ((64*n^2-224*n+164)*x^2-(1024*n^3-5888*n^2+11072*n-6800)*x^3+(2048*n^3-12032*n^2+23040*n-14256)*x^4)*d^2/dx^2 P(n-2,x) - (16*n-36-(512*n^3-3200*n^2+6624*n-4536)*x)*P(n-1,x) + ((64*n^2-256*n+244)*x-(512*n^3-2944*n^2+5472*n-3240)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 56*n + 45 - (512*n^3 - 3200*n^2 + 6112*n - 3256)*x + (9216*n^4 - 79360*n^3 + 251648*n^2 - 347040*n + 174636)*x^2)*P[n - 2, x] - ((64*n^2 - 224*n + 156)*x - (512*n^3 - 2432*n^2 + 3808*n - 2248)*x^2 - (4096*n^4 - 41984*n^3 + 151296*n^2 - 229824*n + 124416)*x^3)*D[P[n - 2, x], x] - ((64*n^2 - 224*n + 164)*x^2 - (1024*n^3 - 5888*n^2 + 11072*n - 6800)*x^3 + (2048*n^3 - 12032*n^2 + 23040*n - 14256)*x^4)*D[P[n - 2, x], {x, 2}] - (16*n - 36 - (512*n^3 - 3200*n^2 + 6624*n - 4536)*x)*P[n - 1, x] + ((64*n^2 - 256*n + 244)*x - (512*n^3 - 2944*n^2 + 5472*n - 3240)*x^2)*D[P[n - 1, x], x])/(16*n^2 - 72*n + 81)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = ((16*n^2 - 56*n + 45 - (512*n^3 - 3200*n^2 + 6112*n - 3256)*x + (9216*n^4 - 79360*n^3 + 251648*n^2 - 347040*n + 174636)*x^2)*v[i - 2] - ((64*n^2 - 224*n + 156)*x - (512*n^3 - 2432*n^2 + 3808*n - 2248)*x^2 - (4096*n^4 - 41984*n^3 + 151296*n^2 - 229824*n + 124416)*x^3)*deriv(v[i - 2], x) - ((64*n^2 - 224*n + 164)*x^2 - (1024*n^3 - 5888*n^2 + 11072*n - 6800)*x^3 + (2048*n^3 - 12032*n^2 + 23040*n - 14256)*x^4)*deriv(deriv(v[i - 2], x), x) - (16*n - 36 - (512*n^3 - 3200*n^2 + 6624*n - 4536)*x)*v[i - 1] + ((64*n^2 - 256*n + 244)*x - (512*n^3 - 2944*n^2 + 5472*n - 3240)*x^2)*deriv(v[i - 1], x))/(16*n^2 - 72*n + 81)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-945ed98594b7abe2","steps":["(\u03b8+1)\u00b2"],"edges":["e-3bb79e8069e7d5c6"],"text":"A290315 \u2192 (\u03b8+1)\u00b2"},"headline":"A290315 \u2192 (\u03b8+1)\u00b2","chain":null}