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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":"eb64f504dfa2558c155b543be115cc3ba90d443a7464b4e765ce1c7248630bf3","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-35bd700768cac55f","role":"sequence","name":"Triangle read by rows: triangle of repeated stepped binomial coefficients.","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: 0, 4;\n  2: 0, 4, 9;\n  3: 0, 0, 9, 16;\n  4: 0, 0, 9, 32, 25;\n  5: 0, 0, 0, 16, 50,  36;\n  6: 0, 0, 0, 16, 75, 108,  49;\n  7: 0, 0, 0,  0, 25, 108, 147,  64;\n  8: 0, 0, 0,  0, 25, 144, 294, 256,  81;\n  9: 0, 0, 0,  0,  0,  36, 196, 384, 324, 100;\n","recurrence_sentence":"The row polynomials satisfy (n^4+6*n^3+13*n^2+12*n+4+(8*n^4+32*n^3+22*n^2-10*n-4)*x+(24*n^4+48*n^3-28*n^2-72*n+16)*x^2+(32*n^4-56*n^2)*x^3+(16*n^4-32*n^3+16*n^2)*x^4)*P(n,x) = -(n^5+5*n^4+7*n^3-n^2-8*n-4+(10*n^5+45*n^4+38*n^3-71*n^2-122*n-44)*x+(40*n^5+144*n^4-8*n^3-402*n^2-178*n+212)*x^2+(80*n^5+184*n^4-296*n^3-536*n^2+444*n+172)*x^3+(80*n^5+48*n^4-384*n^3+120*n^2+200*n+32)*x^4+(32*n^5-48*n^4-32*n^3+64*n^2+16*n-32)*x^5)*P(n-1,x) + ((2*n^4+12*n^3+26*n^2+24*n+8)*x+(18*n^4+92*n^3+134*n^2+20*n-56)*x^2+(64*n^4+256*n^3+180*n^2-196*n-96)*x^3+(112*n^4+304*n^3-40*n^2-256*n-24)*x^4+(96*n^4+128*n^3-144*n^2-64*n+24)*x^5+(32*n^4-32*n^2+16)*x^6)*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[Together[(-(n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 + (10*n^5 + 45*n^4 + 38*n^3 - 71*n^2 - 122*n - 44)*x + (40*n^5 + 144*n^4 - 8*n^3 - 402*n^2 - 178*n + 212)*x^2 + (80*n^5 + 184*n^4 - 296*n^3 - 536*n^2 + 444*n + 172)*x^3 + (80*n^5 + 48*n^4 - 384*n^3 + 120*n^2 + 200*n + 32)*x^4 + (32*n^5 - 48*n^4 - 32*n^3 + 64*n^2 + 16*n - 32)*x^5)*P[n - 1, x] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x + (18*n^4 + 92*n^3 + 134*n^2 + 20*n - 56)*x^2 + (64*n^4 + 256*n^3 + 180*n^2 - 196*n - 96)*x^3 + (112*n^4 + 304*n^3 - 40*n^2 - 256*n - 24)*x^4 + (96*n^4 + 128*n^3 - 144*n^2 - 64*n + 24)*x^5 + (32*n^4 - 32*n^2 + 16)*x^6)*D[P[n - 1, x], x])/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (8*n^4 + 32*n^3 + 22*n^2 - 10*n - 4)*x + (24*n^4 + 48*n^3 - 28*n^2 - 72*n + 16)*x^2 + (32*n^4 - 56*n^2)*x^3 + (16*n^4 - 32*n^3 + 16*n^2)*x^4)]];\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] = (-(n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 + (10*n^5 + 45*n^4 + 38*n^3 - 71*n^2 - 122*n - 44)*x + (40*n^5 + 144*n^4 - 8*n^3 - 402*n^2 - 178*n + 212)*x^2 + (80*n^5 + 184*n^4 - 296*n^3 - 536*n^2 + 444*n + 172)*x^3 + (80*n^5 + 48*n^4 - 384*n^3 + 120*n^2 + 200*n + 32)*x^4 + (32*n^5 - 48*n^4 - 32*n^3 + 64*n^2 + 16*n - 32)*x^5)*v[i - 1] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x + (18*n^4 + 92*n^3 + 134*n^2 + 20*n - 56)*x^2 + (64*n^4 + 256*n^3 + 180*n^2 - 196*n - 96)*x^3 + (112*n^4 + 304*n^3 - 40*n^2 - 256*n - 24)*x^4 + (96*n^4 + 128*n^3 - 144*n^2 - 64*n + 24)*x^5 + (32*n^4 - 32*n^2 + 16)*x^6)*deriv(v[i - 1], x))/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (8*n^4 + 32*n^3 + 22*n^2 - 10*n - 4)*x + (24*n^4 + 48*n^3 - 28*n^2 - 72*n + 16)*x^2 + (32*n^4 - 56*n^2)*x^3 + (16*n^4 - 32*n^3 + 16*n^2)*x^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-35bd700768cac55f","steps":["(\u03b8+1)\u00b2"],"edges":["e-71ab500f4221c0be"],"text":"A103633 \u2192 (\u03b8+1)\u00b2"},"headline":"A103633 \u2192 (\u03b8+1)\u00b2","chain":null}